Chapter 9: Polar Coordinates and Complex Numbers
UNIT 3 Advanced Functions and Graphing
You are now ready to apply what you have learned in earlier units to more complex functions. The three chapters in this unit contain very different topics; however, there are some similarities among them. The most striking similarity is that all three chapters require that you have had some experience with graphing. As you work through this unit, try to make connections between what you have already studied and what you are currently studying. This will help you use the skills you have already mastered more effectively. Chapter 9 Polar Coordinates and Complex Numbers Chapter 10 Conics Chapter 11 Exponential and Logarithmic Functions
550 Unit 3 Advanced Functions and Graphing ORL W D
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I B D E E W • Unit 3 Projects SPACE—THE FINAL FRONTIER People have been fascinated by space since the beginning of time. Until 1961, however, human beings have been bound to Earth, unable to feel and experience life in space. Current space programs undertaken by NASA are exploring our solar system and beyond using sophisticated unmanned satellites such as the Hubble Space Telescope in orbit around Earth and the Mars Global Surveyor in orbit around Mars. In these projects, you will look at some interesting mathematics related to space—the final frontier. At the end of each chapter in Unit 3, you will be given specific tasks to explore space using the Internet.
CHAPTER 9 From Point A to Point B In Chapter 9, you will learn about the polar (page 611) coordinate system, which is quite different from the rectangular coordinate system. Do scientists use the rectangular coordinate system or the polar coordinate system as they record the position of objects in space? Or, do they use some other system? Math Connection: Research coordinate systems by using the Internet. Write a summary that describes each coordinate system that you find. Include diagrams and any information about converting between systems.
CHAPTER 10 Out in Orbit! What types of orbits do planets, artificial satellites, or (page 691) space exploration vehicles have? Can orbits be modeled by the conic sections? Math Connection: Use the Internet to find data about the orbit of a space vehicle, satellite, or planet. Make a scale drawing of the object’s orbit labeling important features and dimensions. Then, write a summary describing the orbit of the object, being sure to discuss which conic section best models the orbit. CHAPTER 11 Kepler is Still King! Johannes Kepler (1571–1630) was an important (page 753) mathematician and scientist of his time. He observed the planets and stars and developed laws for the motion of those bodies. His laws are still used today. It is truly amazing how accurate his laws are considering the primitive observation tools that he used. Math Connection: Research Kepler’s Laws by using the Internet. Kepler’s Third Law relates the distance of planets from the sun and the period of each planet. Use the Internet to find the distance each planet is from the sun and to find each planet’s period. Verify Kepler’s Third Law. •
For more information on the Unit Project, visit: www.amc.glencoe.com
Unit 3 Internet Project 551 Chapter 9 Unit 3 Advanced Functions and Graphing (Chapters 9–11)
POLAR COORDINATES AND COMPLEX NUMBERS
CHAPTER OBJECTIVES • Graph polar equations. (Lessons 9-1, 9-2, 9-4) • Convert between polar and rectangular coordinates. (Lessons 9-3, 9-4) • Add, subtract, multiply, and divide complex numbers in rectangular and polar forms. (Lessons 9-5, 9-7) • Convert between rectangular and polar forms of complex numbers. (Lesson 9-6) • Find powers and roots of complex numbers. (Lesson 9-8)
552 Chapter 9 Polar Coordinates and Complex Numbers 9-1 Polar Coordinates W al or OBJECTIVES e ld R SURVEYING Before large road construction projects, or even the • Graph points A construction of a new home, take place, a surveyor maps out in polar p n o coordinates. plic ati characteristics of the land. A surveyor uses a device called a theodolite • Graph simple to measure angles. The precise locations of various land features are determined polar equations. using distances and the angles measured with the theodolite. While mapping out a • Determine level site, a surveyor identifies a landmark 450 feet away and 30° to the left and the distance between two another landmark 600 feet away and 50° to the right. What is the distance between points with polar the two landmarks? This problem will be solved in Example 5. coordinates.
Recording the position of an object using the distance from a fixed point and an angle made with a fixed ray from that point uses a polar coordinate system. When surveyors record the locations of objects using distances and angles, they are using polar coordinates. In a polar coordinate 105 90 û û 75 7 5 û system, a fixed point O is 120û 2 2 12 12 60û called the pole or origin. 135û 3 3 3 45û The polar axis is usually a 4 4 150 horizontal ray directed toward û5 P(r, ) 6 6 30û the right from the pole. The r 165 11 location of a point P in the û 12 15û 12 polar coordinate system can be 180 0 0 identified by polar coordinates û 1234 û O Polar Axis in the form (r, ). If a ray is 13 23 195û 12 12 345û drawn from the pole through 7 11 point P, the distance from the 210û 6 6 330û pole to point P is r. The 5 7 measure of the angle formed by 4 4 225û 4 5 315û OP and the polar axis is . The 3 17 19 3 240 3 û 12 2 12 300û angle can be measured in 255û 270û 285û degrees or radians. This grid is sometimes called the polar plane.
Consider positive and negative values of r.
Suppose r 0. Then is the measure Suppose r 0. Then is the measure of any angle in standard position that of any angle that has the ray opposite has OP as its terminal side. OP as its terminal side.
P(r, )
O O P(r, )
Lesson 9-1 Polar Coordinates 553 Example 1 Graph each point. a. P(3, 60°) 105û 90û 75û 120û 60û On a polar plane, sketch the 135û 45û terminal side of a 60° angle in P standard position. 150û 30û Since r is positive (r 3), find 165û 15û 60û the point on the terminal side of 180û 0 the angle that is 3 units from the O 1234û pole. 195 û 345û Notice that point P is on the third 210û 330 circle from the pole. û 225û 315û 240û 300û 255û 270û 285û 7 7 5 b. Q 1.5, 2 6 2 12 12 3 3 3 Sketch the terminal side of an 4 4 7 angle measuring radians in 5 6 6 7 6 standard position. 11 6 12 Q 12 Since r is negative, extend the 0 terminal side of the angle in the 1234 opposite direction. Find the O 13 23 point Q that is 1.5 units from the 12 12 pole along this extended ray. 7 11 6 6 Notice that point Q is halfway 5 7 4 4 between the first and second 4 5 3 circles from the pole. 17 3 19 3 12 2 12
As you have seen, the r-coordinate can be any real value. The angle can also be negative. If 0, then is measured counterclockwise from the polar axis. If 0, then is measured clockwise from the polar axis.
Example 2 Graph R( 2, 135°). 105û 90û 75û 120û 60û Negative angles are measured 135û 45û clockwise. Sketch the terminal 150 30û side of an angle of 135° in û R standard position. 165û 15û Since r is negative, the point 180 O 0 R( 2, 135°) is 2 units from the û 1234 û pole along the ray opposite the 195û 345û terminal side that is already 135û drawn. 210û 330û 225û 315û 240û 300û 255û 270û 285û
554 Chapter 9 Polar Coordinates and Complex Numbers In Example 2, the point R( 2, 135°) lies in the polar plane 2 units from the pole on the terminal side of a 45° angle in standard position. This means that point R could also be represented by the coordinates (2, 45°). In general, the polar coordinates of a point are not unique. Every point can be represented by infinitely many pairs of polar coordinates. This happens because any angle in standard position is coterminal with infinitely many other angles. If a point has polar coordinates (r, ), then it also has polar coordinates (r, 2 ) in radians, or (r, 360°) in degrees. In fact, you can add any integer multiple of 2 to and find another pair of polar coordinates for the same point. If you use the opposite r-value, the angle will change by , giving ( r, ) as another ordered pair for the same point. You can then find even more polar coordinates for the same point by adding multiples of 2 to . The following graphs illustrate six of the different ways to name the polar coordinates of the same point.
P P P P 2 120û 480û 420û Polar Axis O O Polar Axis O O Polar Axis Polar Axis 240 60 û 300 û (2, 480 ) û û (2, 120û) or (2, 240û) ( 2, 420û) ( 2, 60û) or ( 2, 300û)
Here is a summary of all the ways to represent a point in polar coordinates.
Multiple If a point P has polar coordinates (r, ), then P can also be represented Representations by polar coordinates (r, 2 k) or ( r, (2k 1) ), where k is of (r, ) any integer.
In degrees, the representations are (r, 360k°) and ( r, (2k 1)180°). For every angle, there are infinitely many representations.
Example 3 Name four different pairs of polar coordinates that represent point S on the graph with the restriction that 360° 360°.
One pair of polar coordinates for 105û 90û 75û point S is (3, 150°). 120û 60û 135û 45û To find another representation, r k k 150û 30û use ( , 360 °) with 1. S (3, 150° 360( 1)°) (3, 210°) 165û 15û To find additional polar coordinates, 180 0 use ( r, (2k 1)180°). û O 1234 û ( 3, 150° 180°) ( 3, 330°) k 0 195û 345û To find a fourth pair, use 210 330 ( r, (2k 1)180°) with k 1. û û ( 3, 150° ( 1)180°) ( 3, 30°) 225û 315û 240û 300û 255û 270û 285û Therefore, (3, 150°), (3, 210°), ( 3, 330°), and ( 3, 30°) all represent the same point in the polar plane.
Lesson 9-1 Polar Coordinates 555 An equation expressed in terms of polar coordinates is called a polar equation. For example, r 2 sin is a polar equation. A polar graph is the set of all points whose coordinates (r, ) satisfy a given polar equation. You already know how to graph equations in the Cartesian, or rectangular, coordinate system. Graphs of equations involving constants like x 2 and y 3 are considered basic in the Cartesian coordinate system. Similarly, the polar coordinate system has some basic graphs. Graphs of the polar equations r k and k, where k is a constant, are considered basic.
Example 4 Graph each polar equation. a. r 3 7 5 The solutions to r 3 are the 2 2 12 12 ordered pairs (r, ) where r 3 3 3 3 and is any real number. Some 4 r 3 4 5 examples are (3, 0), 3, , and 3, 4 6 ()4 6 (3, ). In other words, the graph 11 12 12 of this equation is the set of all (3, ) (3, 0) ordered pairs of the form (3, ). 0 O 1234 Any point that is 3 units from 13 23 the pole is included in this 12 12 graph. The graph is the circle 7 11 centered at the origin with 6 6 5 7 radius 3. 4 4 4 5 3 17 3 19 3 12 2 12 3 b. 4
The solutions to this equation 7 5 are the ordered pairs (r, ) 12 2 12 3 2 where and r is any real 3 3 4 3 number. Some examples are 4 4 5 3 3 3 3 6 3, 6 ( 4 ) 3 1, , 2, , and 3, . 4 4 4 4 11 In other words, the graph of this 12 3 12 ()1, 4 equation is the set of all ordered 0 O 1234 3 pairs of the form r, . The 13 23 3 4 12 ()2, 4 12 graph is the line that includes 7 11 6 6 the terminal side of the angle 3 5 7 in standard position. 4 4 4 4 5 3 17 3 19 3 12 2 12
Just as you can find the distance between two points in a rectangular coordinate plane, you can derive a formula for the distance between two points in a polar plane.
556 Chapter 9 Polar Coordinates and Complex Numbers 7 5 2 Given two points P1(r1, 1) and P2(r2, 2) in the polar 2 12 12 3 3 plane, draw P1OP2. P1OP2 has measure 2 1. Apply 3 the Law of Cosines to P OP . 4 1 2 4 P2(r2, 2) 5 2 2 2 6 6 (P1P2) (OP1) (OP2) 2(OP1)(OP2) cos ( 2 1) 11 2 P1(r1, 1) 12 12 Now substitute r1 for OP1 and r2 for OP2. 1 2 2 2 0 (P1P2) r1 r2 2r1r2 cos ( 2 1) O 1234 2 2 13 23 P1P2 r1 r2 2r1r2 cos ( 2 1) 12 12 7 11 6 6 5 7 4 4 4 5 3 17 3 19 3 12 2 12
Distance If P1(r1, 1) and P2(r2, 2) are two points in the polar plane, then Formula in 2 2 Polar Plane P1P2 r1 r2 2r1r2 cos ( 2 1).
Example 5 SURVEYING Refer to the application at the beginning of the lesson. What is W the distance between the two landmarks? al or e ld R Set up a coordinate system so that turning to the A p n left is a positive angle and turning to the right is p o L L2 lic ati a negative angle. With this convention, the 1 landmarks are at L (450, 30°) and L (600, 50°). 30û 1 2 50û 450 600
Now use the Polar Distance Formula. O
2 2 P1P2 r1 r2 2r1r2 cos ( 2 1) (r1, 1) (450, 30°) 2 2 L1L2 450 600 2(450)(600) cos ( 50° 30°) (r2, 2) (600, 50°) 684.6 feet The landmarks are about 685 feet apart.
C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Explain why a point in the polar plane cannot be named by a unique ordered pair (r, ).
2. Explain how to graph (r, ) if r 0 and 0.
3. Name two values of such that ( 4, ) represents the same point as (4, 120°).
4. Explain why the graph in Example 4a is also the graph of r 3.
5. Describe the polar coordinates of the pole.
Lesson 9-1 Polar Coordinates 557 Guided Practice Graph each point. 13 6. A(1, 135°) 7. B 2.5, 8. C( 3, 120°) 9. D 2, 6 6
10. Name four different pairs of polar coordinates that represent the point at 2, . 6 Graph each polar equation. 11. r 1 12. 13. r 3.5 3 14. Find the distance between P 2.5, and P 3, on the polar plane. 1 6 2 4 15. Gardening A lawn sprinkler can cover the part of a circular region determined by the polar inequalities 30° 210° and 0 r 20, where r is measured in feet. a. Sketch a graph of the region that the sprinkler can cover. b. Find the area of the region.
E XERCISES
Practice Graph each point. 1 3 A 16. E(2, 30°) 17. F 1, 18. G(5, 240°) 19. H , 2 2 4 5 20. J 1.5, 21. K , 210° 22. L 3, 23. M(2, 90°) 4 2 6 11 25 7 24. N( 1, 120°) 25. P 0.5, 26. Q 2, 27. R , 1050° B 6 3 2 7 5 28. List four pairs of polar coordinates 2 2 12 12 that represent the point S in the 3 3 3 graph. Use both radians and 4 4 degrees. 5 6 S 6 11 12 12
0 1234 O 13 23 12 12 7 11 6 6 5 7 4 4 4 5 3 17 3 19 3 Name four other pairs of polar coordinates 12 2 12 for each point. 29. T(1.5, 180°) 30. U 1, 31. V(4, 315°) 3 Graph each polar equation. 5 32. r 1.5 33. 34. r 2 4 35. 30° 36. 150° 37. 4 C 38. 840° 39. r 0 40. r 1
558 Chapter 9 Polar Coordinates and Complex Numbers www.amc.glencoe.com/self_check_quiz 41. Write a polar equation for the circle centered at the origin with radius 2 .
Find the distance between the points with the given polar coordinates.
42. P1(4, 170°) and P2(6, 105°) 3 43. P 1, and P 5, 1 6 2 4 2 44. P 2.5, and P 1.75, 1 8 2 5 45. P1(1.3, 47°) and P2( 3.6, 62°)
46. Find an ordered pair of polar coordinates to represent the point whose rectangular coordinates are ( 3, 4).
Applications 47. Web Page Design When designing websites with circular and Problem graphics, it is often convenient to use polar coordinates, Solving sometimes called “pizza coordinates” in this context. If the W al or origin is at the center of the screen, what are the polar e ld R equations of the lines that cut the region into the six A p n o plic ati congruent slices shown?
48. Critical Thinking Prove that if P1(r1, ) and P2(r2, ) are two points in the polar plane, then the distance formula reduces to P1P2 r1 r2.
O 90 49. Sailing A graph of the maximum speed of a sailboat 5 10 15 20 û versus the angle of the wind is called a “performance curve” or just a “polar.” The graph at the right is the 105
polar for a particular boat when the wind speed is 120 20 knots. represents the angle of the wind in degrees, and r is the maximum speed of the boat in knots. 135 150 a. What is the maximum speed when 120°? 180 165 b. What is the maximum speed when 150°?
50. Acoustics Polar coordinates can be used to model a concert amphitheater. Suppose the performer is placed at the pole and faces the direction of the polar axis. The seats have been built to occupy the region with and 0.25 r 3, 3 3 where r is measured in hundreds of feet. a. Sketch this region in the polar plane. b. How many seats are there if each person has 6 square feet of space?
51. Critical Thinking Explain why the order of the points used in the distance formula is irrelevant. That is, why can you choose either point to be P1 and the other to be P2?
Lesson 9-1 Polar Coordinates 559 52. Aviation Two jets at the same altitude are being tracked on an air traffic controller’s radar screen. The coordinates of the planes are (5, 310°) and (6, 345°), with r measured in miles. a. Sketch a graph of this situation. b. If regulations prohibit jets from passing within three miles of each other, are these planes in violation? Explain.
U.S. Navy Blue Angels
Mixed Review 53. Transportation Two docks are directly across a river from each other. A boat that can travel at a speed of 8 miles per hour in still water is attempting to cross directly from one dock to the other. The current of the river is 3 miles per hour. At what angle should the captain head? (Lesson 8-5)
54. Find the inner product 3, 2, 4 1, 4, 0 . Then state whether the vectors are perpendicular. Write yes or no. (Lesson 8-4)
55. Find the distance from the line with equation y 9x 3 to the point at ( 3, 2). (Lesson 7-7) 1 sin2 56. Simplify the expression . (Lesson 7-1) sin2 3 57. Find Arccos . (Lesson 6-8) 2 58. State the amplitude and period for the function y 5 cos 4 . (Lesson 6-4)
59. Determine the number of possible solutions and, if a solution exists, solve ABC if A 30°, b 18.6, and a 9.3. Round to the nearest tenth. (Lesson 5-7)
60. Find the number of possible positive real zeros and the number of possible negative real zeros of the function f(x) x 3 4x2 4x 1. Then determine the rational zeros. (Lesson 4-4) x 2 2x 3 61. Determine the slant asymptote of f(x) . (Lesson 3-7) x 5 62. Determine whether the graph of f(x) x 4 3x 2 2 is symmetric with respect to the x-axis, the y-axis, both, or neither. (Lesson 3-1) 2 4 1 63. Find the value of the determinant 1 1 0 . (Lesson 2-5) 3 4 5 64. Given that x is an integer, state the relation representing y 11 x and 3 x 0 by listing a set of ordered pairs. Then state whether this relation is a function. (Lesson 1-1)
65. SAT/ACT Practice The circumference of the circle is 50 . What A is the length of the diagonal AB of the square inscribed in the circle? 1 A 12 B 10 2 C 25 2 B D 25 2 E 50
560 Chapter 9 Polar Coordinates and Complex Numbers Extra Practice See p. A42. 9-2 Graphs of Polar Equations W al or OBJECTIVE e ld R AUDIO TECHNOLOGY One way to • Graph polar A describe the ability of a microphone to equations. p n o plic ati pick up sounds from different directions is to examine its polar pattern. A polar coordinate system is set up with the microphone at the origin. is used to locate a source of sound that moves in a horizontal circle around the microphone, and r measures the amplitude of the signal that the microphone detects. The polar graph of r as a function of is called the polar pattern of the microphone. A cardioid microphone is a microphone whose polar pattern is shaped like the graph of the equation r 2.5 2.5 cos . A graph of this polar pattern provides information about the microphone. A problem related to this will be solved in Example 3.
In Lesson 9-1, you learned to graph polar equations of the form r k and k, where k is a constant. In this lesson, you will learn how to graph more complicated types of polar equations. When you graph any type of polar equation for the first time, you can create a table of values for r and . Then you can use the table to plot points in the polar plane.
Example 1 Graph r sin . Make a table of values. Round the values of r to the nearest tenth. Graph the ordered pairs and connect them with a smooth curve. sin (r, ) 105û 90û 75û 120û 60û 0° 0 (0, 0°) 135û 45û 30° 0.5 (0.5, 30°) 30 45° 0.7 (0.7, 45°) 150û û 60° 0.9 (0.9, 60°) 165û 15û 90° 1 (1, 90°) 180 0 120° 0.9 (0.9, 120°) û O 0.25 0.5 0.75 1 û 135° 0.7 (0.7, 135°) 195 345û 150° 0.5 (0.5, 150°) û 180° 0 (0, 180°) 210û 330û 210° 0.5 ( 0.5, 210°) 225û 315û 225° 0.7 ( 0.7, 225°) 240û 300û 240° 0.9 ( 0.9, 240°) 255û 270û 285û 270° 1( 1, 270°) Notice that the ordered pairs 300° 0.9 ( 0.9, 300°) obtained when 180° 360° 315° 0.7 ( 0.7, 315°) represent the same points as the 330° 0.5 ( 0.5, 330°) ordered pairs obtained when 360° 0 (0, 360°) 0° 180°.
Lesson 9-2 Graphs of Polar Equations 561 The graph of r cos is shown 105û 90û 75û 60 at the right. Notice that this graph 120û û and the graph in Example 1 are 135û 45û circles with a diameter of 1 unit. 150û 30û Both pass through the origin. As with the families of graphs you 165û 15û studied in Chapter 3, you can alter 180 0 the position and shape of a polar û O 0.25 0.5 0.75 1 û graph by multiplying the function by 195û 345û a number or by adding to it. You can also multiply by a number or add a 210û 330û number to it in order to alter the 225û 315û graph. However, the changes in 240û 300û the graphs of polar equations can 255û 270û 285û be quite different from those you studied in Chapter 3.
Examples 2 Graph each polar equation. a. r 3 5 cos
7 5 2 3 5 cos (r, ) 2 12 12 3 3 3 0 2(2, 0) 4 4 5 1.3 1.3, 6 6 6 6 11 12 12 0.5 0.5, 3 3 0 O 246 8 3 3, 2 2 13 23 12 12 2 2 5.5 5.5, 7 11 3 3 6 6 5 7 5 5 4 4 7.3 7.3, 4 5 6 6 3 17 3 19 3 12 8(8, ) 2 12 7 7 7.3 7.3, 6 6 4 4 5.5 5.5, 3 3 3 3 3 3, 2 2 5 5 0.5 0.5, 3 3 11 11 1.3 1.3, 6 6 2 2( 2, 2 )
This type of curve is called a limaçon. Some limaçons have inner loops like this one. Other limaçons come to a point, have a dimple, or just curve outward.
562 Chapter 9 Polar Coordinates and Complex Numbers b. r 3 sin 2
3 sin 2 (r, ) 3 sin 2 (r, ) 3 3 00 (0, 0) 0 0, 2 2 2.6 2.6, 5 5 6 6 2.6 2.6, 3 3 3 3, 7 7 4 4 3 3, 4 4 2.6 2.6, 11 11 3 3 2.6 2.6, 6 6 0 0, 2 2 2 2 2.6 2.6, 7 5 3 3 2 2 12 12 3 3 3 3 3 3, 3 4 4 4 4 5 5 5 2.6 2.6, 6 6 6 6 11 0(0. ) 12 12 7 7 2.6 2.6, 0 6 6 O 123 4 5 5 13 23 3 3, 12 12 4 4 7 11 4 4 2.6 2.6, 6 6 3 3 5 7 4 4 4 5 3 This type of curve is called a 17 3 19 3 rose. 12 2 12
Example 3 AUDIO TECHNOLOGY Refer to the application at the beginning of the W al or lesson. The polar pattern of a microphone can be modeled by the polar e ld R equation r 2.5 2.5 cos .
A p n a. Sketch a graph of the polar pattern. p io lic at b. Describe what the polar pattern tells you about the microphone. 90 105û û 75û a. 2.5 2.5 cos (r, ) 120û 60û 0° 5 (5, 0°) 135û 45û 30° 4.7 (4.7, 30°) 150û 30û 60° 3.8 (3.8, 60°) 165û 15û 90° 2.5 (2.5, 90°) 180 120° 1.3 (1.3, 120°) û O 246 80û 150° 0.3 (0.3, 150°) 195û 345û 180° 0 (0, 180°) 210° 0.3 (0.3, 210°) 210û 330û 240° 1.3 (1.3, 240°) 225û 315û 270° 2.5 (2.5, 270°) 240û 300û 255û 270 285û 300° 3.8 (3.8, 300°) û 330° 4.7 (4.7, 330°) (continued on the next page)
Lesson 9-2 Graphs of Polar Equations 563 This type of curve is called a cardioid. A cardioid is a special type of limaçon.
b. The polar pattern indicates that the microphone will pick up only very loud sounds from behind it. It will detect much softer sounds from in front.
Limaçons, roses, and cardioids are examples of classical curves. The classical curves are summarized in the chart below.
Classical Curves spiral of lemniscate limaçon cardioid Archimedes Curve rose (pronounced (pronounced (pronounced (pronounced lehm NIHS kuht) lee muh SOHN) KARD ee oyd) ar kih MEED eez) r a cos n Polar r 2 a2 cos 2 r a b cos r a a cos r a r a sin n Equation r 2 a2 sin 2 r a b sin r a a sin ( in radians) n is a positive integer.
General Graph
It is possible to graph more than one polar equation at a time on a polar plane. However, the points where the graphs intersect do not always represent common solutions to the equations, since every point can be represented by infinitely many polar coordinates.
Example 4 Graph the system of polar equations. Solve the system using algebra and trigonometry and compare the solutions to those on your graph. r 3 3 sin Graphing r 4 sin Calculator To solve the system of equations, Tip substitute 3 3 sin for r in the second A graphing calculator equation. can graph polar 3 3 sin 4 sin equations in its polar mode. You can use the 2 sin 1 simultaneous graphing 1 sin mode to help determine 2 if the graphs pass 7 11 or through the same point 6 6 [ 6, 6] scl:1 by [ 6, 6] scl:1 at the same time. Substituting each angle into either of the original equations gives r 4.5, so 7 11 the solutions of the system are 4.5, and 4.5, . Tracing the curves 6 6 shows that these solutions correspond with the intersection points of the graphs.
564 Chapter 9 Polar Coordinates and Complex Numbers C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Write a polar equation whose graph is a rose. 2. Determine the maximum value of r in the equation r 3 5 sin . What is the minimum value of r? 3. State the reason that algebra and trigonometry do not always find all the points of intersection of the graphs of polar equations. 4. You Decide Linh and Barbara were working on their homework together. Linh said that she thought that when graphing polar equations, you only need to generate points for which 0 because other values of would just generate the same points. Barbara said she thought she remembered an example from class where values of ranging from 0 to 4 had to be considered. Who is correct? Explain.
Guided Practice Graph each polar equation. Identify the type of curve each represents. 5. r 1 sin 6. r 2 3 sin 7. r cos 2 8. r 1.5
9. Graph the system of polar equations r 2 sin and r 2 cos 2 . Solve the system using algebra and trigonometry. Assume 0 2 . 10. Biology The chambered nautilus is a mollusk whose shell can be modeled by the polar equation r 2 . a. Graph this equation for 0 2 . b. Determine an approximate interval for that would result in a graph that models the chambered nautilus shown in the photo.
E XERCISES
Practice Graph each polar equation. Identify the type of curve each represents. A 11. r 3 sin 12. r 3 3 cos 13. r 3 B 14. r2 4 cos 2 15. r 2 sin 3 16. r 2 sin 3 5 17. r 18. r 5 3 cos 19. r 2 2 sin 2 20. r2 9 sin 2 21. r sin 4 22. r 2 2 cos
23. Write an equation for a rose with 3 petals. 24. What is an equation for a spiral of Archimedes that passes through A , ? 4 2
Graph each system of polar equations. Solve the system using algebra and trigonometry. Assume 0 2 . C 25. r 3 26. r 1 cos 27. r 2 sin r 2 cos r 1 cos r 2 sin 2
www.amc.glencoe.com/self_check_quiz Lesson 9-2 Graphs of Polar Equations 565 Graphing Graph each system using a graphing calculator. Find the points of intersection. Calculator Round coordinates to the nearest tenth. Assume 0 2 . 28. r 1 29. r 3 3 sin 30. r 2 2 cos r 2 cos 2 r 2 r 3 sin
Applications 31. Textiles Patterns in fabric can often be created by and Problem modifying a mathematical graph. The pattern at the Solving right can be modeled by a lemniscate. W al or e ld R a. Suppose the designer wanted to begin with a
A p n lemniscate that was 6 units from end to end. pli tio c a What polar equation could have been used?
b. What polar equation could have been used to generate a lemniscate that was 8 units from end to end?
32. Audio Technology Refer to the application at the beginning of the lesson and Example 3. Another microphone has a polar pattern that can be modeled by the polar equation r 3 2 cos . Graph this polar pattern and compare it to the pattern of the cardioid microphone.
33. Music The curled part at the end of a violin is called the scroll. The scroll in the picture appears to curl around twice. For what interval of -values will the graph of r model this violin scroll?
34. Critical Thinking a. Graph r cos for n 2, 4, 6, and 8. Predict the n shape of the graph of r cos . 10 b. Graph r cos for n 3, 5, 7, and 9. Predict the n shape of the graph of r cos . 11
35. Communication Suppose you want to use your computer to make a heart- shaped card for a friend. Write a polar equation that you could use to generate the shape of a heart. Make sure the graph of your equation points in the right direction.
36. Critical Thinking The general form for a limaçon is r a b cos or r a b sin . a. When will a limaçon have an inner loop? b. When will it have a dimple? c. When will it have neither an inner loop nor a dimple? (This is called a convex limaçon.)
566 Chapter 9 Polar Coordinates and Complex Numbers 37. Critical Thinking Describe the transformation necessary to obtain the graph of each equation from the graph of the polar function r f( ). a. r f( ) ( is a constant.) b. r f( ) c. r f( ) d. r cf( ) (c is a constant, c 0.)
Mixed Review 38. Find four other pairs of polar coordinates that represent the same point as (4, 45°). (Lesson 9-1) 39. Find the cross product of v 2, 3, 0 and w 1, 2, 4 . Verify that the resulting vector is perpendicular to v and w. (Lesson 8-4) 40. Use a ruler and protractor to determine the magnitude (in centimeters) and direction of the vector shown at the right. (Lesson 8-1)
x
sin2 x 41. Verify that tan2 x is an identity. (Lesson 7-2) cos4 x cos2 x sin2 x 42. Solve ABC if A 21°15 , B 49°40 , and c 28.9. Round angle measures to the nearest minute and side measures to the nearest tenth. (Lesson 5-6) 43. Travel Adita is trying to decide where to go on vacation. He prefers not to fly, so he wants to take a bus or a train. The table below shows the round-trip fares for trips from his home in Kansas City, Missouri to various cities. Represent this data with a matrix. (Lesson 2-3)
New Los York Angeles Miami Bus $240 $199 $260 Train $254 $322 $426
1 6 8 4 44. SAT Practice Which fraction is the simplified form of ? 3 26 16 A 3 14 B 3 28 C 9 7 D 3 4 E 3
Extra Practice See p. A42. Lesson 9-2 Graphs of Polar Equations 567 9-3 Polar and Rectangular Coordinates W al or OBJECTIVE e ld R PHYSIOLOGY A laboratory has • Convert F A designed a voice articulation program between polar p n o and rectangular plic ati that synthesizes speech by controlling coordinates. the speech articulators: the jaw, tongue, lips, and C J so on. To create a mathematical model that is manageable, each of these speech articulators is identified by a point. J the edge of the jaw C the center of the tongue F a fixed point about which the jaw rotates The positions of these points are given in polar coordinates with F as the pole and the horizontal as the polar axis. Changing these positions alters the sounds that are synthesized. A problem related to this will be solved in Example 2.
For some real-world phenomena, it is useful to be able to convert between polar coordinates and rectangular coordinates.
90 Suppose a rectangular coordinate 105û y û 75û 120û 60û system is superimposed on a polar 135û 45û coordinate system so that the origins 150û r 30û coincide and the x-axis aligns with the 165 15 polar axis. Let P be any point in the plane. û û 180 0 Polar coordinates: P(r, ) û 1 2 3 4 x û O Rectangular coordinates: P(x, y) 195û 345û 210û 330û 225û 315û 240 300 û 255 285 û û 270û û
Trigonometric functions can be used to convert polar coordinates to rectangular coordinates.
Converting The rectangular coordinates (x, y) of a point named by the polar Polar coordinates (r, ) can be found by using the following formulas. Coordinates to Rectangular x r cos Coordinates y r sin You will derive these formulas in Exercise 42.
568 Chapter 9 Polar Coordinates and Complex Numbers Examples 1 Find the rectangular coordinates of each point. a. P 5, 3 For P 5, , r 5 and . 3 3 x r cos y r sin 5 cos 5 sin 3 3 1 5 3 5 3 5 or 5 or 2 2 2 2 5 5 3 The rectangular coordinates of P are , or (2.5, 4.33) to the nearest 2 2 hundredth.
b. Q( 13, 70°) For Q( 13, 70°), r 13 and 70°. x r cos y r sin 13 cos ( 70°) 13 sin ( 70°) 13(0.34202) 13( 0.93969) 4.45 12.22 The rectangular coordinates of Q are approximately ( 4.45, 12.22).
2 PHYSIOLOGY Refer to the application at the beginning of the lesson. W Suppose the computer model assigns polar coordinates (7.5, 330°) to point J al or e ld R and (4.5, 310°) to point C in order to create a particular sound. Each unit
A represents a centimeter. Is the center of the tongue above or below the edge p n o of the jaw? by how far? plic ati Find the rectangular coordinates of each point. y O x For J(7.5, 330°), r 7.5 and 330°. x r cos y r sin (2.89, 3.45) C J 7.5 cos 330° 7.5 sin 330° (6.50, 3.75) 6.50 3.75 J(7.5, 330°) → J(6.50, 3.75)
For C(4.5, 310°), r 4.5 and 310°. x r cos y r sin 4.5 cos 310° 4.5 sin 310° 2.89 3.45 C(4.5, 310°) → C(2.89, 3.45)
Since 3.45 3.75, the center of the tongue is above the edge of the jaw when this sound is made. Subtracting the y-coordinates, we see that the center of the tongue is about 0.3 centimeter, or 3 millimeters, higher than the edge of the jaw.
Lesson 9-3 Polar and Rectangular Coordinates 569 If a point is named by the rectangular coordinates (x, y), you can find the corresponding polar coordinates by using the Pythagorean Theorem and the Arctangent function. Since the Arctangent function only determines angles in the first and fourth quadrants, you must add radians to the value of for points with coordinates (x, y) that lie in the second or third quadrants.
y P(x, y) y
y y Arctan Arctan x x O x O x
P(x, y) y y When x is zero, When x 0, Arctan . When x 0, Arctan . x x . Why? 2
The polar coordinates (r, ) of a point named by the rectangular Converting coordinates (x, y) can be found by the following formulas. Rectangular r x 2 y 2 Coordinates y to Polar Arctan , when x 0 Coordinates x y Arctan , when x 0 x
Example 3 Find the polar coordinates of R( 8, 12). For R( 8, 12), x 8 and y 12. y r x 2 y 2 Arctan x 0 x 2 2 ( 8) ( 12) 1 2 Arctan 208 8 3 Arctan 14.42 2 4.12 The polar coordinates of R are approximately (14.42, 4.12). Other polar coordinates can also represent this point.
The conversion equations can also be used to convert equations from one coordinate system to the other.
Examples 4 Write the polar equation r 6 cos in rectangular form. r 6 cos r2 6r cos Multiply each side by r. x2 y2 6x r2 x2 y2, rcos x
570 Chapter 9 Polar Coordinates and Complex Numbers Example 5 Write the rectangular equation (x 3)2 y2 9 in polar form. (x 3)2 y2 9 (r cos 3)2 (r sin )2 9 x r cos , y r sin r2 cos2 6r cos 9 r2 sin2 9 Multiply. r2 cos2 6r cos r2 sin2 0 Subtract 9 from each side. r2 cos2 r2 sin2 6r cos Isolate squared terms. r2 (cos2 sin2 ) 6r cos Factor. r2(1) 6r cos Pythagorean Identity r2 6r cos r 6 cos Simplify.
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Communicating Read and study the lesson to answer each question. Mathematics 1. Write the polar coordinates of the point in the graph at the y right.
2. Explain why you have to consider what quadrant a point O x lies in when converting from rectangular coordinates to polar coordinates.
3. Determine the polar equation for x 2.
4. Math Journal Write a paragraph explaining how to convert from polar coordinates to rectangular coordinates and vice versa. Include a diagram with your explanation.
Guided Practice Find the polar coordinates of each point with the given rectangular coordinates. Use 0 2 and r 0. 5. 2 , 2 6. ( 2, 5)
Find the rectangular coordinates of each point with the given polar coordinates. 4 7. 2, 8. (2.5, 250°) 3 Write each rectangular equation in polar form. 9. y 2 10. x2 y2 16
Write each polar equation in rectangular form. 11. r 6 12. r sec
13. Acoustics The polar pattern of a cardioid lavalier microphone is given by r 2 2 cos . a. Graph the polar pattern. b. Will the microphone detect a sound that originates from the point with rectangular coordinates ( 2, 0)? Explain.
www.amc.glencoe.com/self_check_quiz Lesson 9-3 Polar and Rectangular Coordinates 571 E XERCISES A Practice Find the polar coordinates of each point with the given rectangular coordinates. Use 0 2 and r 0. A 14. (2, 2) 15. (0, 1) 16. 1, 3 1 3 B 17. , 18. (3, 8) 19. (4, 7) 4 4 Find the rectangular coordinates of each point with the given polar coordinates. 1 3 20. 3, 21. , 22. 1, 2 2 4 6 23. ( 2, 270°) 24. (4, 210°) 25. (14, 130°)
Write each rectangular equation in polar form. 26. x 7 27. y 5 28. x2 y2 25 29. x2 y2 2y 30. x2 y2 1 31. x2 (y 2)2 4
Write each polar equation in rectangular form. 32. r 2 33. r 3 C 34. 35. r 2 csc 3 36. r 3 cos 37. r2 sin 2 8
38. Write the equation y x in polar form. 39. What is the rectangular form of r sin ?
Applications 40. Surveying A surveyor identifies a landmark at the point with polar and Problem coordinates (325, 70°). What are the rectangular coordinates of this Solving point?
l Wo a rl e d 41. Machinery An arc of a spiral of Archimedes R
A is used to create a disc that drives the p n o plic ati spindle on a sewing machine. Note from the figure that the spinning disc drives the rod that moves the piston. The outline of the disc can be modeled by the graph of 5 r for and its reflection 6 4 4 in the line . How far does the piston 4 move from right to left?
42. Critical Thinking Write a convincing argument to prove that, when converting polar coordinates to rectangular coordinates, the formulas x r cos and y r sin are true. Include a labeled drawing in your answer.
572 Chapter 9 Polar Coordinates and Complex Numbers y 43. Irrigation A sod farm can use a combined Cartesian and 5 polar coordinate system to identify points in the field. Points 4 have coordinates of the form (x, y, r, ) . The sprinkler heads 3 2 are spaced 25 meters apart. The Cartesian coordinates x and 1 y, which are positive integers, indicate how many sprinkler x heads to go across and up in the grid. The polar coordinates O 1 2345 give the location of the point relative to the sprinkler head. If a point has coordinates (4, 3, 2, 120°), then how far to the east and north of the origin is that point?
44. Electrical Engineering When adding sinusoidal expressions like 4 sin (3.14t 20°) and 5 sin (3.14t 70°), electrical engineers sometimes use phasors, which are vectors in polar form. a. The phasors for these two expressions are written as 4 20° and 5 70°. Find the rectangular forms of these vectors. b. Add the two vectors you found in part a. c. Write the sum of the two vectors in phasor notation. d. Write the sinusoidal expression corresponding to the phasor in part c.
45. Critical Thinking Identify the type of graph generated by the polar equation r 2a sin 2a cos . Write the equivalent rectangular equation.
Mixed Review 46. Graph the polar equation r 2 sin . (Lesson 9-2)
47. Name four different pairs of polar coordinates that represent the point at ( 2, 45°). (Lesson 9-1)
48. Aviation An airplane flies at an air speed of 425 mph on a heading due south. It flies against a headwind of 50 mph from a direction 30° east of south. Find the airplane’s ground speed and direction. (Lesson 8-5)
49. Solve sin2 A cos A 1 for principal values of A. (Lesson 7-5)
50. Graph y 2 cos . (Lesson 6-4)
51. Use the unit circle to find the exact value of cos 210°. (Lesson 5-3)
52. Write a polynomial function to model the set of data. (Lesson 4-8)
x 10 7 4 1 2581114 f(x) 15 9.2 6.9 3 0.1 2 1.1 2.3 4.5
53. Use synthetic division to divide x5 3x2 20 by x 2. (Lesson 4-3)
54. Write the equation of best fit for a set of data using the ordered pairs (17, 145) and (25, 625). (Lesson 1-6) x y 55. SAT/ACT Practice x, y, and z are different positive integers. and are also y z positive integers. Which of the following cannot be a positive integer? x z A B (x)(y) C D (x y)z E (x z)y z x
Extra Practice See p. A42. Lesson 9-3 Polar and Rectangular Coordinates 573 9-4 Polar Form of a Linear Equation W al or OBJECTIVES e ld R BIOLOGY Scientists hope that by studying the behavior of flies, they • Write the polar A will gain insight into the genetics and brain functions of flies. Karl Götz form of a linear p n o equation. plic ati of the Max Planck Institute in Tübingen, Germany, designed the Buridan • Graph the polar Paradigm, which is a device that tracks the path of a fruit fly walking between two form of a linear visual landmarks. The position equation. MONITOR OBJECT FINDER of the fly is recorded in both POSITION PULSE SECTOR rectangular and polar X Y P1 P2 S coordinates. If the fly walks in a straight line from COMPUTER the point with polar coordinates (6, 2) to the point with polar coordinates (2, 0.5), what is the equation of the path of the fly? How close did the fly come to the origin? This problem will be solved in Example 4.
The polar form of the equation for a y Look Back You can refer to line is closely related to the normal form, Lesson 7-6 to review which is x cos y sin p 0. We P(r, ) normal form. have learned that x r cos and y r sin . The polar form of the equation of line can be obtained by substituting these values into the normal form. p x cos y sin p 0 (r cos ) cos (r sin ) sin p 0 O x r(cos cos sin sin ) p r cos ( ) p cos cos sin sin cos ( )
The polar form of a linear equation, where p is the length of the normal Polar Form of a Linear and is the positive angle between the positive x-axis and the normal, is Equation p r cos ( ).
In the polar form of a linear equation, and r are variables, and p and are constants. Values for p and can be obtained from the normal form of the standard equation of a line. Remember to choose the value for according to the quadrant in which the normal lies.
574 Chapter 9 Polar Coordinates and Complex Numbers Example 1 Write each equation in polar form. a. 5x 12y 26 The standard form of the equation is 5x 12y 26 0. First, write the equation in normal form to find the values of p and . To convert to normal form, we need to find the value of A 2 + B 2.
A 2 B 2 5 2 12 2, or 13
Since C is negative, use 13. The normal form of the equation is 5 12 x y 2 0. The normal form is x cos y sin p 0. 13 13 5 12 We can see from the normal form that p 2, cos , and sin . 13 13 Since cos and sin are both positive, the normal lies in the first quadrant. sin tan cos 1 2 tan 13 5 13 12 tan 5 1.18 Use the Arctangent function. Substitute the values for p and into the polar form. p r cos ( ) → 2 r cos ( 1.18) The polar form of 5x 12y 26 is 2 r cos ( 1.18).
b. 2x 7y 5 The standard form of this equation is 2x 7y 5 0. So, A 2 B 2 22 ( 7)2, or 53 . Since C is positive, use 53 . Then the normal form of the equation is 2 7 5 x y 0. 53 53 53
5 5 53 2 7 We see that p or , cos , and sin . 53 53 53 53 Since cos 0 but sin 0, the normal lies in the second quadrant. sin tan cos 7 7 2 7 53 tan 2 53 53 53 2 106° Add 180° to the Arctangent value. Substitute the values for p and into the polar form. 5 53 p r cos ( ) → r cos ( 106°) 53 5 53 The polar form of the equation is r cos ( 106°). 53
Lesson 9-4 Polar Form of a Linear Equation 575 The polar form of a linear equation can be converted to rectangular form by using the angle sum and difference identities for cosine and the polar coordinate conversion equations.
Example 2 Write 2 r cos ( 60°) in rectangular form. 2 r cos ( 60°) 2 r(cos cos 60° sin sin 60°) Difference identity for cosine 1 3 1 3 2 r cos sin cos 60° , sin 60° 2 2 2 2 1 3 2 r cos r sin Distributive Property 2 2 1 3 2 x y Polar to rectangular conversion equations 2 2 4 x 3 y Multiply each side by 2. 0 x 3 y 4 Subtract 4 from each side. The rectangular form of 2 r cos ( 60°) is x 3 y 4 0.
The polar form of a linear equation can be graphed by preparing a table of coordinates and then graphing the ordered pairs in the polar system.
2 Graph the equation r 4 sec . Examples 3 3
4 Graphing Write the equation in the form r . Calculator 2 cos Tip 3 Use your calculator to make a table of values. You can use the TABLE feature with Tbl 2 5 to generate these 0 6 6 3 2 3 6 values. r 8 4.6 4 4.6 8 undefined 8
Graph the ordered pairs on a polar plane. 7 5 2 2 12 12 3 3 3 4 4 5 6 6 11 12 12
0 O 246 8 13 23 12 12 7 11 6 6 5 7 4 4 4 5 3 17 3 19 3 12 2 12
576 Chapter 9 Polar Coordinates and Complex Numbers Example 4 BIOLOGY Refer to the application at the beginning of the lesson. W al or e ld a. If the fly walks in a straight line from the point with polar coordinates R (6, 2) to the point with polar coordinates (2, 0.5), what is the equation of A p n the path of the fly? o plic ati b. If r is measured in centimeters, how close did the fly come to the origin? a. The two points must satisfy the equation p r cos ( ). Substitute both ordered pairs into this form to create a system of equations. p 6 cos (2 ) (r, ) (6, 2) p 2 cos ( 0.5 ) (r, ) (2, 0.5) Substituting either expression for p into the other equation results in the equation 6 cos (2 ) 2 cos ( 0.5 ). A graphing calculator shows that there are two solutions to this equation y 6cos(2 X ) between 0 and 2 : 0.59 and 3.73. Substituting these values into p 6 cos (2 ) yields p 0.93 and y 2cos( 0.5 X ) p 0.93, respectively. Since p, the length of the normal, must be positive, we use 0.59 and p 0.93. [0, 2 ] scl:1 by [ 6, 6] scl:1 Therefore, the polar form of the equation of the fly’s path is 0.93 r cos ( 0.59). b. The closest that the fly came to the origin is the value of p, 0.93 centimeter or 9.3 millimeters.
C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. State the general form for the polar equation of a line. Explain the significance of each part of the equation. 2. Determine what value of will result in r p in the polar equation of a line. 3. Explain how to find and the length of the normal for a rectangular equation of the form x k. Write the polar form of the equation. 4. Explain why it is a good idea to find several ordered pairs when graphing the polar equation of a line, even though only two points are needed to determine a line.
Guided Practice Write each equation in polar form. Round to the nearest degree. 5. 3x 4y 10 0 6. 2x 4y 9
Write each equation in rectangular form. 7. 3 r cos( 60°) 8. r 2 sec 4
Lesson 9-4 Polar Form of a Linear Equation 577 Graph each polar equation. 9. 3 r cos 10. r 2 sec ( 45°) 3
11. Aviation An air traffic controller is looking at a radar screen with the control tower at the origin. 5 a. If the path of an airplane can be modeled by the equation 5 r cos , 6 then what are the polar coordinates of the plane when it comes the closest to the tower? b. Sketch a graph of the path of the plane.
E XERCISES
Practice Write each equation in polar form. Round to the nearest degree. A 12. 7x 24y 100 0 13. 21x 20y 87 14. 6x 8y 21 15. 3x 2y 5 0 B 16. 4x 5y 10 17. x 3y 7
Write each equation in rectangular form. 18. 6 r cos ( 120°) 19. 4 r cos 4 20. 2 r cos ( ) 21. 1 r cos ( 330°) 7 22. r 11 sec 23. r 5 sec ( 60°) 6 Graph each polar equation. 24. 6 r cos ( 45°) 25. 1 r cos 6 26. 2 r cos ( 60°) 27. 3 r cos ( 90°) C 28. r 3 sec 29. r 4 sec 3 4
30. Write the polar form of the equation of the line that passes through points with rectangular coordinates (4, 1) and ( 2, 3). 31. What is the polar form of the equation of the line that passes through points 7 with polar coordinates 3, and 2, ? 4 6
Applications 32. Biology Refer to the application at the and Problem beginning of the lesson. Suppose the Solving Buridan Paradigm tracks a fruit fly whose W al or path is modeled by the polar equation e ld R r 6 sec ( 15°), where r is measured A p n o plic ati in centimeters. a. How close did the fly come to the origin? b. What were the polar coordinates of the fly when it was closest to the origin? 33. Critical Thinking Write the polar forms of the equations of two lines, neither of which is vertical, such that the lines intersect at a 90° angle and have normal segments of length 2.
578 Chapter 9 Polar Coordinates and Complex Numbers www.amc.glencoe.com/self_check_quiz 34. Robotics The diagram at the right shows y a robot with a telescoping arm. The “hand” at the end of such an arm is called the R manipulator. What polar equation should the Manipulator arm be programmed to follow in order to move the manipulator in a straight line from the point with rectangular coordinates (5, 4) x to the point with rectangular coordinates (15, 4)? 35. Surveying A surveyor records the locations of points in a plot of land by means of polar coordinates. In a circular plot of radius 500 feet, stakes are placed at (125, 130°) and (300, 70°), where r is measured in feet. The stakes are at the same elevation. a. Draw a diagram of the plot of land. Include the locations of the two stakes. b. Find the polar equation of the line determined by the stakes. 36. Critical Thinking Show that k r sin ( ) is also the equation of a line in polar coordinates. Identify the significance of k and in the graph. 37. Design A carnival ride designer wants to create a Ferris wheel with a 40-foot radius. The designer wants the interior of the circle to have lines of lights that form a regular pentagon. Find the polar equation of the line that contains (40, 0°) and (40, 72°).
Mixed Review 38. Write the polar equation r 6 in rectangular form. (Lesson 9-3) 39. Identify the type of curve represented by the equation r sin 6 . (Lesson 9-2) 40. Write parametric equations of the line with equation x 3y 6. (Lesson 8-6) 41. Gardening A sprinkler is set to rotate 65° and spray a distance of 6 feet. What is the area of the gound being watered? (Lesson 6-1) 42. Use the unit circle to find the value of sin 360° (Lesson 5-3) 43. Solve 2x3 5x2 12x 0. (Lesson 4-1) 44. SAT Practice Grid-In If c d 12 and c2 d 2 48, then c d ?
MID-CHAPTER QUIZ
Graph each polar equation. (Lesson 9-1) 7. Write the polar form of the equation 2 2 2 x y 36. (Lesson 9-3) 1. r 4 2. 3
8. Find the rectangular form of r 2 csc . Graph each polar equation. (Lesson 9-2) (Lesson 9-3) 3. r 3 3 sin 4. r cos 2
Write each equation in polar form. Round Find the polar coordinates of each point with to the nearest degree. (Lesson 9-4) the given rectangular coordinates. (Lesson 9-3) 9. 5x 12y 3 5. P( 2 , 2 ) 6. Q(0, 4) 10. 2x 6y 2
Extra Practice See p. A42. Lesson 9-4 Polar Form of a Linear Equation 579 9-5 Simplifying Complex Numbers W al or OBJECTIVE e ld R DYNAMICAL SYSTEMS • Add, subtract, 11,326 A Dynamical systems is a branch multiply, and p n o divide complex plic ati of mathematics that studies 10,896 numbers in constantly changing systems like the stock rectangular form. market, the weather, and population. In many 10,466 cases, one can catch a glimpse of the system 5/25 8/25 at some point in time, but the forces that act A Typical Graph of the Stock Market on the system cause it to change quickly. By analyzing how a dynamical system changes over time, it may be possible to predict the behavior of the system in the future. One of the basic mathematical models of a dynamical system is iteration of a complex function. A problem related to this will be solved in Example 4.
Recall that complex numbers are numbers of the form a bi, where a and b are real numbers and i, the imaginary unit, is defined by i 2 1. The first few powers of i are shown below.
i 1 ii2 1 i 3 i 2 i ii4 (i 2)2 1 i 5 i 4 i ii6 i 4 i 2 1 i 7 i 4 i 3 ii8 (i 2)4 1
Notice the repeating pattern of the powers of i. To find the value of i n, i, 1, i, 1, i, 1, i, 1 let R be the remainder In general, the value of in, where n is a whole when n is divided by 4. number, can be found by dividing n by 4 and if R 0 i n 1 examining the remainder as summarized in the if R 1 i n i table at the right. if R 2 i n 1 You can also simplify any integral power of i by if R 3 i n i rewriting the exponent as a multiple of 4 plus a positive remainder.
Example 1 Simplify each power of i. a. i53 b. i 13 Method 1 Method 2 Method 1 Method 2 53 4 13 R1 i 53 (i 4)13 i 13 4 4 R3 i 13 (i 4) 4 i 3 If R 1, i n i. (1)13 i If R 3, i n i. (1) 4 i 3 i 53 i ii 13 i i
580 Chapter 9 Polar Coordinates and Complex Numbers 581 imaginary part. . i (continued on the next page) 1 0, the complex number is an number is 0, the complex ] i 4 is called the is called 2 , find the first five iterates of b 2 i are real numbers, is said to be in is said to numbers, real are z b Distributive property i Distributive property ) i , then the complex number is a complex number , then the i b i i 3 6 i ) 0.25 ) and 0.5 1 Lesson 9-5 Simplifying Complex Numbers ) [ i and i 2 3 1 i i a 14 0.5 z Replace z with 1 i 12( 12 0, as in 4 2)] i 0.25 2 (0.5 i (2 4 i 0.5 i . Describe any pattern that you see. 2 eal part ( i i i 0.5 4 0.5 r i 8 8 b i 0.5 ) , where , where 4 i z ) ( b )( ) 1 i 10 f i 0.5 i i [5 3 i i 0.25 i 3 3 If ). 21 21 0 and z 0.5 i ) Complex numbers can be added and subtracted by added and subtracted numbers can be Complex a 0 29 0.5 ) ) ) i 4 i z i i 6 )(1 for the independent variable. 6 2 4 4 i a z z i 7(2 14 2 14 0.5 0.25 z is called the is called If i 0.5 ) (0.5 ) i 2 )(7 0.5 2 2 a i i i i ) (14 (14 3 i 0.5 ( ( 0.5 ) 4 0.5 0.5 i ) ) is the process of repeatedly applying a function to the output of repeatedly is the process (0.5 i 0.5 i i ) ) 0.5 i i 2 i 2 0.5 3 3 )(7 ) (0.5 (0.5 0.5 i i 3 0.5 0, the complex number is a real number. If number. a real number is 0, the complex ) (5 (10 ) i z for the initial value The complex number number complex The Iteration The product of two or more complex numbers can be found using the same of two or more The product ( ( (1 ( f Simplify each expression. a. (5 DYNAMICAL SYSTEMS f f f f b. (10 Simplify (2 (2 b ectangular form.ectangular
pure imaginary number. imaginary and real on both the the chosen operation performing parts. imaginary number. r If produced by the previous input. When using complex numbers with functions, input. When using complex numbers with functions, by the previous produced it is traditional to use procedures you use when multiplying binomials. procedures 4 3 2
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e p R A p Graphing Calculator Tip Example Example Example Some calculators have a complex number mode. In this mode, they can perform complex number arithmetic. . i b , and i complex 0.25 is the i i . ) i 2 i i ) ) 2 bc d d 0.25 are called are i b b ad b ( ( 2 ( i 0.5, i i 1 ) ) a ) 2 c c , ) , the following are true. i 0.125 ad d i i 0.25 bd d rite the answer in the form a and 0.5 2 a a i Multiply by 1; 1 conjugate of 1 i W i 2 ( ( 0.25 b bc c c 2 ac 2 ( i i 0.25 ) ) 0.5 0.125 a i i 6 1 ) d d 2 i i and 1 2 2 , 0.25 ) bd d 2 i 0.25 i i i i 3 i b 3 d 4) 4 4 6( c c z 1 1 )( ( ( 2 i are are ( i i ac c i i 7 5 5 1 a i 0.125 c i i 7 ). i i 7 z 10 i 0.5 3 2 3 2 ) ) )( 1 2 5 i i i 0.5) i i b b b 1 i 5 i 5 b d 5 5 1 11 1 5 1 )( (1 i (0.5 0.25 0.125 a a a 0.25 c a ( ( ( ) i 0.5 ) 2 ) i i 3
0.25 (1 0.25 (0.5 Recall that if a quadratic equation with real coefficients has complex with real Recall that if a quadratic equation ) . The absolute values of the nonzero real and imaginary real parts of the nonzero . The absolute values (1, 0.5, ) i i 3 0.5 0.25 wo complex numbers of the form wo complex numbers 0.25 T The list below summarizes the operations with complex numbers presented ( ( halved. for two steps and then are 0.25) stay the same (5 The first five iterates of 1 The first five iterates For any complex numbers f f Simplify (5 in this lesson. conjugates. conjugates also complex conjugates. Complex solutions, then those solutions are simplify the quotient To in the division of complex numbers. play a useful role the numerator and denominator by the of two complex numbers, multiply is similar to rationalizing the The process conjugate of the denominator. like denominator in an expression . 5 Example Chapter 9 Polar Coordinates and Complex Numbers Numbers Operations download a download
aphing with Complex 582 Graphing Calculator Programs To gr calculator that program performs complex visit: iteration, www.amc. glencoe.com C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Describe how to simplify any integral power of i. 2. Draw a Venn diagram to show the relationship between real, pure imaginary, and complex numbers. 3. Explain why it is useful to multiply by the conjugate of the denominator over itself when simplifying a fraction containing complex numbers. 4. Write a quadratic equation that has two complex conjugate solutions.
Guided Practice Simplify. 5. i 6 6. i 10 i 2 7. (2 3i) ( 6 i) 8. (2.3 4.1i) ( 1.2 6.3i) 9. (2 4i) ( 1 5i) i 10. ( 2 i)2 11. 1 2i Look Back 12. Vectors It is sometimes convenient to use complex numbers to represent You can refer to vectors. A vector with a horizontal component of magnitude a and a vertical Lessons 8-1 and 8-2 component of magnitude b can be represented by the complex number a bi. to review vectors, If an object experiences a force with a horizontal component of 2.5 N and a components, and vertical component of 3.1 N as well as a second force with a horizontal resultants. component of 6.2 N and a vertical component of 4.3 N, find the resultant force on the object. Write your answer as a complex number.
E XERCISES
Practice Simplify. A 13. i 6 14. i 19 15. i 1776 16. i 9 i 5 17. (3 2i) ( 4 6i) 18. (7 4i) (2 3i) 1 19. i (2 i) 20. ( 3 i) (4 5i) 2 B 21. (2 i)(4 3i) 22. (1 4i)2
23. 1 7 i 2 5 i 24. 2 3 1 12 2 i 3 2i 5 i 25. 26. 27. 1 2i 4 i 5 i
C 28. Write a quadratic equation with solutions i and i. 29. Write a quadratic equation with solutions 2 i and 2 i.
Simplify. 30. (2 i)(3 2i)(1 4i) 31. ( 1 3i)(2 2i)(1 2i) 1 3 i 2 2 2 i 32. 33. 1 2 i 3 6 i 3 i (1 i)2 34. 35. (2 i)2 ( 3 2i)2
www.amc.glencoe.com/self_check_quiz Lesson 9-5 Simplifying Complex Numbers 583 Applications 36. Electricity Impedance is a measure of how much hindrance there is to the flow and Problem of charge in a circuit with alternating current. The impedance Z depends on the Solving resistance R, the reactance due to capacitance XC, and the reactance due to W al or inductance X in the circuit. The impedance is written as the complex number e ld L R Z R (X X )j. (Electrical engineers use j to denote the imaginary unit.) A L C p n o plic ati In the first part of a particular series circuit, the resistance is 10 ohms, the reactance due to capacitance is 2 ohms, and the reactance due to inductance is 1 ohm. In the second part of the circuit, the respective values are 3 ohms, 1 ohm, and 1 ohm. a. Write complex numbers that represent the impedances in the two parts of the circuit. b. Add your answers from part a to find the total impedance in the circuit. c. The admittance of an AC circuit is a measure of how well the circuit allows current to flow. Admittance is the reciprocal 1 of impedance. That is, S . The units for admittance are Z siemens. Find the admittance in a circuit with an impedance of 6 3j ohms.
37. Critical Thinking a. Solve the equation x2 8ix 25 0. b. Are the solutions complex conjugates? c. How does your result in part b compare with what you already know about complex solutions to quadratic equations? d. Check your solutions.
38. Critical Thinking Sometimes it is useful to separate a complex function into its real and imaginary parts. Substitute z x yi into the function f(z) z2 to write the equation of the function in terms of x and y only. Simplify your answer.
39. Dynamical Systems Find the first five iterates for the given function and initial value. a. f(z) iz, z0 2 i b. f(z) (0.5 0.866i)z, z0 1 0i 40. Critical Thinking Simplify (1 2i) 3. 41. Physics One way to derive the equation of motion in a spring-mass system is to solve a differential equation. The solutions of such a differential equation typically involve expressions of the form cos t i sin t. You generally expect solutions that are real numbers in such a situation, so you must use algebra to eliminate the imaginary numbers. Find a relationship between the constants c1 and c2 such that c1(cos 2t i sin 2t) c2(cos 2t i sin 2t) is a real number for all values of t.
Mixed Review 42. Write the equation 6x 2y 3 in polar form. (Lesson 9-4) 43. Graph the polar equation r 4 . (Lesson 9-2) 44. Write a vector equation of the line that passes through P( 3, 6) and is parallel to v 1, 4 . (Lesson 8-6)
584 Chapter 9 Polar Coordinates and Complex Numbers 1 45. Find an ordered triple to represent u if u v 2 w, v 8, 6, 4 , and 4 w 2, 6, 3 . (Lesson 8-3)
46. If and are measures of two first quadrant angles, find cos ( ) if 4 5 tan and cot . (Lesson 7-3) 3 12 47. A twig floats on the water, bobbing up and down. The distance between its highest and lowest points is 7 centimeters. It moves from its highest point down to its lowest point and back up to its highest point every 12 seconds. Write a cosine function that models the movement of the twig in relationship to the equilibrium point. (Lesson 6-6)
48. Surveying A surveyor finds that the angle of elevation from a certain point to the top of a cliff is 60°. From a point 45 feet farther away, the angle of elevation to the top of the cliff is 52°. How high is the cliff to the nearest foot? (Lesson 5-4) 49. What type of polynomial function would be the best model for the set of data? (Lesson 4-8)
x 3 1 1357911 f(x) 4 2 3861 3 8
50. Construction A community wants to build a second pool at their community park. Their original pool has a width 5 times its depth and a length 10 times its depth. They wish to make the second pool larger by increasing the width of the original pool by 4 feet, increasing the length by 6 feet, and increasing the depth by 2 feet. The volume of the new pool will be 3420 cubic feet. Find the dimensions of the original pool. (Lesson 4-4) 51. If y varies jointly as x and z and y 80 when x 5 and z 8, find y when x 16 and z 2. (Lesson 3-8) 52. If f(x) 7 x2, find f 1(x). (Lesson 3-4) 53. Find the maximum and minimum values of the function f(x, y) 2x y for the polygonal convex set determined by the system of inequalities. (Lesson 2-6) x 6 y 1 y x 2 54. Solve the system of equations. (Lesson 2-2) x 2y 7z 14 x 3y 5z 21 5x y 2z 7
55. SAT/ACT Practice If BC BD in the figure, D what is the value of x 40? xû A 100 B 80 C 60 D 40 E cannot be determined from the information 120û given ACB
Extra Practice See p. A43. Lesson 9-5 Simplifying Complex Numbers 585 9-6 The Complex Plane and Polar Form of Complex Numbers W al or OBJECTIVES e ld R FRACTALS One of the standard ways to generate a fractal involves • Graph complex 2 A iteration of a quadratic function. If the function f(z) z is iterated numbers in the p n o complex plane. plic ati using a complex number as the initial input, there are three possible • Convert complex outcomes. The terms of the sequence of outputs, called the orbit, may numbers from rectangular to • increase in absolute value, polar form and • decrease toward 0 in absolute value, or vice versa. • always have an absolute value of 1. One way to analyze the behavior of the orbit is to graph the numbers in the complex plane. Plot the first five members of the orbit of z0 0.9 0.3i under iteration by f(z) z2. This problem will be solved in Example 3.
Recall that a bi is referred to as the rectangular form of a complex number. The rectangular form is sometimes written as an ordered pair, (a, b). Two complex numbers in rectangular form are equal if and only if their real parts are equal and their imaginary parts are equal.
Example 1 Solve the equation 2x y 3i 9 xi yi for x and y, where x and y are real numbers. 2x y 3i 9 xi yi (2x y) 3i 9 (x y)i On each side of the equation, group the real parts and the imaginary parts. 2x y 9 and x y 3 Set the corresponding parts equal to each other. x 4 and y 1 Solve the system of equations.
Complex numbers can be graphed in the imaginary (i ) complex plane. The complex plane has a real axis and an imaginary axis. The real axis is horizontal, and the imaginary axis is vertical. The complex number a bi is graphed as the ordered pair (a, b) in the complex O real ( ) plane. The complex plane is sometimes called the Argand plane.
Recall that the absolute value of a real number is its i a bi distance from zero on the number line. Similarly, the absolute value of a complex number is its distance from b zero in the complex plane. When a bi is graphed in the O a complex plane, the distance from zero can be calculated using the Pythagorean Theorem.
586 Chapter 9 Polar Coordinates and Complex Numbers Absolute Value of a Complex If z a bi, thenz a 2 b 2. Number
Examples 2 Graph each number in the complex plane and find its absolute value. a. z 3 2i b. z 4i
i i (0, 4) (3, 2)
O O
z 3 2i z 0 4i z 3 2 2 2 z 0 2 4 2 13 4
Example 3 FRACTALS Refer to the application at the beginning of the lesson. Plot l Wo the first five members of the orbit of z 0.9 0.3i under iteration by a rl 0 e d 2 R f(z) z .
A p n First, calculate the first five members of the orbit. Round the real and p o lic ati imaginary parts to the nearest hundredth. z1 0.72 0.54i z1 f(z0) z2 0.23 0.78i z2 f(z1) z3 0.55 0.35i z3 f(z2) z 0.18 0.39i z f(z ) 4 4 3 i 1 z5 0.12 0.14i z5 f(z4) z2 z Then graph the numbers in the complex plane. 1 z3 The iterates approach the origin, so their absolute O values decrease toward 0. 1 1 z5 z4
1
So far we have associated the complex number a bi i with the rectangular coordinates (a, b). You know from Lesson 9-1 that there are also polar coordinates (r, ) associated with the same point. In the case of a complex r number, r represents the absolute value, or modulus, of the b complex number. The angle is called the amplitude or O a argument of the complex number. Since is not unique, it may be replaced by 2 k, where k is any integer.
Lesson 9-6 The Complex Plane and Polar Form of Complex Numbers 587 As with other rectangular coordinates, complex coordinates can be written in polar form by substituting a r cos and b r sin . z a bi r cos (r sin )i r (cos i sin ) This form of a complex number is often called the polar or trigonometric form.
Polar Form of The polar form or trigonometric form of the complex number a bi is a Complex Number r (cos i sin ). r (cos i sin ) is often abbreviated as r cis .
Values for r and can be found by using the same process you used when changing rectangular coordinates to polar coordinates. For a bi, r a 2 b 2 and Arctan b if a 0 or Arctan b if a 0. The amplitude is usually a a expressed in radian measure, and the angle is in standard position along the polar axis.
Example 4 Express each complex number in polar form. a. 3 4i
First, plot the number in the complex plane. ( 3, 4) i Then find the modulus. r ( 3)2 42 or 5 Now find the amplitude. Notice that is in Quadrant II. O 4 Arctan 3 2.21 Therefore, 3 4i 5(cos 2.21 i sin 2.21) or 5 cis 2.21.
b. 1 3 i i First, plot the number in the complex plane. 2 ( 1, 3 ) Then find the modulus. 1 r 12 3 2 or 2 O 221 1 Now find the amplitude. Notice that is in 1 Quadrant I. 2 3 Arctan or 1 3 Therefore, 1 3 i 2 cos i sin or 2 cis . 3 3 3
588 Chapter 9 Polar Coordinates and Complex Numbers You can also graph complex numbers in polar form.
11 11 Graph 4 cos i sin . Then express it in rectangular form. Example 5 6 6 In the polar form of this complex 7 5 12 2 12 2 i number, the value of r is 4, and the 3 3 11 3 value of is . Plot the point 4 6 4 11 5 with polar coordinates 4, . 6 6 6 11 To express the number in 12 12 rectangular form, simplify the 0 1234 trigonometric values: O 11 11 4 cos i sin 13 23 6 6 12 12 3 1 7 11 4 i 6 6 2 2 5 7 4 4 4 5 2 3 2i 3 17 3 19 3 12 2 12
C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Explain how to find the absolute value of a complex number. 2. Write the polar form of i. 3. Find a counterexample to the statement z1 z2 z1 z2 for all complex numbers z1 and z2. 4. Math Journal Your friend is studying complex numbers at another school at the same time that you are. She learned that the absolute value of a complex number is the square root of the product of the number and its conjugate. You know that this is not how you learned it. Write a letter to your friend explaining why this method gives the same answer as the method you know. Use algebra, but also include some numerical examples of both techniques.
Guided Practice 5. Solve the equation 2x y xi yi 5 4i for x and y, where x and y are real numbers.
Graph each number in the complex plane and find its absolute value. 6. 2 i 7. 1 2 i
Express each complex number in polar form. 8. 2 2i 9. 4 5i 10. 2
Graph each complex number. Then express it in rectangular form. 3 11. 4 cos i sin 12. 2(cos 3 i sin 3) 13. (cos 2 i sin 2 ) 3 3 2 14. Graph the first five members of the orbit of z0 0.25 0.75i under iteration by f(z) z2 0.5.
Lesson 9-6 The Complex Plane and Polar Form of Complex Numbers 589 15. Vectors The force on an object is represented by the complex number 10 15i, where the components are measured in newtons. a. What is the magnitude of the force? b. What is the direction of the force?
E XERCISES
Practice Solve each equation for x and y, where x and y are real numbers. A 16. 2x 5yi 12 15i 17. 1 (x y)i y 3xi 18. 4x yi 5i 2x y xi 7i
Graph each number in the complex plane and find its absolute value. 19. 2 3i 20. 3 4i 21. 1 5i 22. 3i 23. 1 5 i 24. 4 2 i
25. Find the modulus of z 4 6i.
Express each complex number in polar form. B 26. 3 3i 27. 1 3 i 28. 6 8i 29. 4 i 30. 20 21i 31. 2 4i 32. 3 33. 4 2 34. 2i
Graph each complex number. Then express it in rectangular form. 35. 3 cos i sin 36. cos i sin 4 4 6 6 4 4 37. 2 cos i sin 38. 10(cos 6 i sin 6) 3 3 5 5 39. 2 cos i sin 40. 2.5(cos 1 i sin 1) 4 4 41. 5(cos 0 i sin 0) 42. 3(cos i sin )
Graph the first five members of the orbit of each initial value under iteration by the given function. 2 2 C 43. z 0.5 i, f(z) z2 0.5 44. z i, f(z) z2 0 0 2 2 2 45. Graph the first five iterates of z0 0.5 0.5i under f(z) z 0.5.
Applications 46. Electrical Engineering Refer to Exercise 44 in Lesson 9-3. Consider a and Problem circuit with alternating current that contains two voltage sources in series. Solving Suppose these two voltages are given by v1(t) 40 sin (250t 30°) and W al or v (t) 60 sin (250t 60°), where t represents time, in seconds. e ld 2 R a. The phasors for these two voltage sources are written as 40 30° A p n o plic ati and 60 60°, respectively. Convert these phasors to complex numbers in rectangular form. (Use j as the imaginary unit, as electrical engineers do.) b. Add these two complex numbers to find the total voltage in the circuit. c. Write a sinusoidal function that gives the total voltage in the circuit.
590 Chapter 9 Polar Coordinates and Complex Numbers www.amc.glencoe.com/self_check_quiz 47. Critical Thinking How are the polar forms of complex conjugates alike? How are they different?
48. Electricity A series circuit contains two sources of impedance, one of 10(cos 0.7 j sin 0.7) ohms and the other of 16(cos 0.5 j sin 0.5) ohms. a. Convert these complex numbers to rectangular form. b. Add your answers from part a to find the total impedance in the circuit. c. Convert the total impedance back to polar form.
49. Transformations Certain operations with complex numbers correspond to geometric transformations in the complex plane. Describe the transformation applied to point z to obtain point w in the complex plane for each of the following operations. a. w z (2 3i) b. w i z c. w 3z d. w is the conjugate of z
50. Critical Thinking Choose any two complex numbers, z1 and z2, in rectangular form.
a. Find the product z1z2.
b. Write z1, z2, and z1z2 in polar form. c. Repeat this procedure with a different pair of complex numbers. d. Make a conjecture about the product of two complex numbers in polar form.
Mixed Review 51. Simplify (6 2i)( 2 3i). (Lesson 9-5)
52. Find the rectangular coordinates of the point with polar coordinates ( 3, 135°). (Lesson 9-3)
53. Find the magnitude of the vector 3,7 , and write the vector as a sum of unit vectors. (Lesson 8-2)
54. Use a sum or difference identity to find tan 105°. (Lesson 7-3)
55. Mechanics A pulley of radius 18 centimeters turns at 12 revolutions per second. What is the linear velocity of the belt driving the pulley in meters per second? (Lesson 6-2)
56. If a 12 and c 18 in ABC, find the measure of B angle A to the nearest tenth of a degree. (Lesson 5-5) c a 57. Solve 2 a 1 3 a 5. (Lesson 4-7) A b C 58. Without graphing, describe the end behavior of the graph of y 2x2 2. (Lesson 3-5)
59. SAT/ACT Practice A person is hired for a job that pays $500 per month and receives a 10% raise in each following month. In the fourth month, how much will that person earn? A $550 B $600.50 C $650.50 D $665.50 E $700
Extra Practice See p. A43. Lesson 9-6 The Complex Plane and Polar Form of Complex Numbers 591 GRAPHING CALCULATOR EXPLORATION 9-6B Geometry in the
OBJECTIVE Complex Plane • Explore geometric An Extension of Lesson 9-6 relationships in the complex Many geometric figures and relationships can be described by using complex plane. numbers. To show points on figures, you can store the real and imaginary parts of the complex numbers that correspond to the points in lists L1 and L2 and use STAT PLOT to graph the points.
TRY THESE 1. Store 1 2i as M and 1 5i as N. Now consider complex numbers of the form (1 T)M TN, where T is a real number. You can generate several numbers of this form and store their real and imaginary parts in L1 and L2, respectively, by entering the following instructions on the home screen. seq( is in the LIST OPS menu. real( and imag( are in the MATH CPX menu.
Use a graphing window of [ 10, 10] sc1:1 by [ 25, 25] sc1:5. Turn on Plot 1 and use a scatter plot to display the points defined in L1 and L2. What do you notice about the points in the scatter plot? 2. Are the original numbers M and N shown in the scatter plot? Explain. 3. Repeat Exercise 1 storing 1 1.5i as M and 2 i as N. Describe your results. 4. Repeat Exercises 1 and 2 for several complex numbers M and N of your choice. (You may need to change the window settings.) Then make a conjecture about where points of the form (1 T)M TN are located in relation to M and N. 5. Suppose K, M, and N are three noncollinear points in the complex plane. Where will you find all the points that can be expressed in the form aK bM cN, where a, b, and c are nonnegative real numbers such that a b c 1? Use the calculator to check your answer.
WHAT DO YOU 6. In Exercises 1-4, where is (1 T)M TN in relation to M and N if the value of THINK? T is between 0 and 1? 7. Where in the complex plane will you find the complex numbers z that satisfy the equation z (1 i) 5? 8. What equation models the points in the complex plane that lie on the circle of radius 2 that is centered at the point 2 3i?
592 Chapter 9 Polar Coordinates and Complex Numbers 9-7 Products and Quotients of Complex Numbers in Polar Form W al or OBJECTIVE e ld R ELECTRICITY Complex numbers can be used in the study of electricity, • Find the product A specifically alternating current (AC). There are three basic quantities to and quotient p n o of complex plic ati consider: numbers in • the current I, measured in amperes, polar form. • the impedance Z to the current, measured in ohms, and • the electromotive force E or voltage, measured in volts. These three quantities are related by the equation E I Z. Current, impedance, and voltage can be expressed as complex numbers. Electrical engineers use j as the imaginary unit, so they write complex numbers in the form a bj. For the total impedance a bj, the real part a represents the opposition to current flow due to resistors, and the imaginary part b is related to the opposition due to inductors and capacitors. If a circuit has a total impedance of 2 6j ohms and a voltage of 120 volts, find the current in the circuit. This problem will be solved in Example 3.
Multiplication and division of complex numbers in polar form are closely tied to geometric transformations in the complex plane. Let r1(cos 1 i sin 1) and r2(cos 2 i sin 2) be two complex numbers in polar form. A formula for the product of the two numbers can be derived by multiplying the two numbers directly and simplifying the result. r1(cos 1 i sin 1) r2(cos 2 i sin 2) 2 r1r2(cos 1 cos 2 i cos 1 sin 2 i sin 1 cos 2 i sin 1 sin 2) 2 r1r2[(cos 1 cos 2 sin 1 sin 2) i(sin 1 cos 2 cos 1 sin 2)] i 1 r1r2[cos ( 1 2) i sin ( 1 2)] Sum identities for cosine and sine
Product of Complex r1(cos 1 i sin 1) r2(cos 2 i sin 2) Numbers in r1r2[cos ( 1 2) i sin ( 1 2)] Polar Form
Lesson 9-7 Products and Quotients of Complex Numbers in Polar Form 593 Notice that the modulus (r1r2) of the product of the two complex numbers is the product of their moduli. The amplitude ( 1 2) of the product is the sum of the amplitudes.
7 7 2 2 Find the product 3 cos i sin 2 cos i sin . Then express Example 1 6 6 3 3 the product in rectangular form. Find the modulus and amplitude of the product. r r1r2 1 2 7 2 3(2) 6 3 11 6 6
11 11 The product is 6 cos i sin . 6 6 Now find the rectangular form of the product.
11 11 3 1 11 3 11 1 6 cos i sin 6 i cos , sin 6 6 2 2 6 2 6 2 3 3 3i The rectangular form of the product is 3 3 3i.
Suppose the quotient of two complex numbers is expressed as a fraction. A formula for this quotient can be derived by rationalizing the denominator. To rationalize the denominator, multiply both the numerator and denominator by the same value so that the resulting new denominator does not contain imaginary numbers.
r (cos i sin ) 1 1 1 r2(cos 2 i sin 2) r (cos i sin ) (cos i sin ) 1 1 1 2 2 cos 2 i sin 2 is the r (cos i sin ) (cos i sin ) 2 2 2 2 2 conjugate of cos 2 i sin 2. r (cos cos sin sin ) i(sin cos cos sin ) 1 1 2 1 2 1 2 1 2 2 2 r2 cos 2 sin 2 r 1 [cos ( 1 2) i sin ( 1 2)] Trigonometric identities r2
Quotient of Complex r (cos i sin ) r 1 1 1 1 [cos ( ) i sin ( )] r 1 2 1 2 Numbers in r2(cos 2 i sin 2) 2 Polar Form
r Notice that the modulus 1 of the quotient of two complex numbers is the r2 quotient of their moduli. The amplitude ( 1 2) of the quotient is the difference of the amplitudes.
594 Chapter 9 Polar Coordinates and Complex Numbers 3 3 Example 2 Find the quotient 12 cos i sin 4 cos i sin . Then express 4 4 2 2 the quotient in rectangular form. Find the modulus and amplitude of the quotient. r 1 r 1 2 r2 1 2 3 4 4 2 5 3 4 5 5 The quotient is 3 cos i sin . 4 4 Now find the rectangular form of the quotient. 5 2 5 5 cos , 3 cos i sin 3 2 2 i 4 2 4 4 2 2 5 sin 2 4 2 3 2 3 2 i 2 2 The rectangular form of the quotient is 3 2 3 2 i. 2 2
You can use products and quotients of complex numbers in polar form to solve the problem presented at the beginning of the lesson.
Example 3 ELECTRICITY If a circuit has an impedance of 2 6j ohms and a voltage of W 120 volts, find the current in the circuit. al or e ld R Express each complex number in polar form. A p n o 120 120(cos 0 j sin 0) plic ati 2 6j 40 [cos ( 1.25) j sin ( 1.25)] r 2 2 ( 6)2 40 or 2 10 , 6 2 10 [cos ( 1.25) j sin ( 1.25)] Arctan or 1.25 2
Substitute the voltage and impedance into the equation E I Z. E I Z 120(cos 0 j sin 0) I 2 10 [cos ( 1.25) j sin ( 1.25)] 120(cos 0 j sin 0) I 2 10 [cos ( 1.25) j sin ( 1.25)]
6 10 (cos 1.25 j sin 1.25) I Now express the current in rectangular form. I 6 10 (cos 1.25 j sin 1.25) 5.98 18.01j Use a calculator. 6 10 cos 1.25 5.98, 6 10 sin 1.25 18.01 The current is about 6 18j amps.
Lesson 9-7 Products and Quotients of Complex Numbers in Polar Form 595 C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Explain how to find the quotient of two complex numbers in polar form. 2. Describe how to square a complex number in polar form. 3. List which operations with complex numbers you think are easier in rectangular form and which you think are easier in polar form. Defend your choices with examples.
Guided Practice Find each product or quotient. Express the result in rectangular form. 3 3 4. 2 cos i sin 2 cos i sin 2 2 2 2 2 2 5. 3 cos i sin 4 cos i sin 6 6 3 3 9 9 6. 4 cos i sin 2 cos i sin 4 4 2 2 1 5 5 7. cos i sin 6 cos i sin 2 3 3 6 6
8. Use polar form to find the product 2 2 3 i 3 3 i . Express the result in rectangular form.
9. Electricity Determine the voltage in a circuit when there is a current of 11 11 2 cos j sin amps and an impedance of 3 cos j sin ohms. 6 6 3 3
E XERCISES
Practice Find each product or quotient. Express the result in rectangular form. 2 2 10. 4 cos i sin 7 cos i sin A 3 3 3 3 3 3 11. 6 cos i sin 2 cos i sin 4 4 4 4 1 B 12. cos i sin 3 cos i sin 2 3 3 6 6 3 3 13. 5(cos i sin ) 2 cos i sin 4 4 5 5 14. 6 cos i sin 3 cos i sin 3 3 6 6 7 7 15. 3 cos i sin cos i sin 3 3 2 2 16. 2(cos 240° i sin 240°) 3(cos 60° i sin 60°) 7 7 2 3 3 17. 2 cos i sin cos i sin 4 4 2 4 4 18. 3(cos 4 i sin 4) 0.5(cos 2.5 i sin 2.5)
596 Chapter 9 Polar Coordinates and Complex Numbers www.amc.glencoe.com/self_check_quiz 19. 4[cos ( 2) i sin ( 2)] (cos 3.6 i sin 3.6) 7 7 11 11 20. 20 cos i sin 15 cos i sin 6 6 3 3 3 3 21. 2 cos i sin 2 cos i sin 4 4 2 2
22. Find the product of 2 cos i sin and 6 cos i sin . Write the 3 3 6 6 answer in rectangular form. z 5 5 1 1 23. If z1 4 cos i sin and z2 cos i sin , find and express 3 3 2 3 3 z2 the result in rectangular form.
Use polar form to find each product or quotient. Express the result in rectangular form. C 24. (2 2i) ( 3 3i) 25. 2 2 i 3 2 3 2 i
26. 3 i 2 2 3 i 27. 4 2 4 2 i (6 6i)
Applications 28. Electricity Find the current in a circuit with a voltage of 13 volts and an and Problem impedance of 3 2j ohms. Solving 29. Electricity Find the impedance in a circuit with a voltage of 100 volts and a W al or e ld current of 4 3j amps. R
A p n o 7 5 plic ati 30. Critical Thinking Given z and z 2 1 2 2 12 i 12 graphed at the right, graph z z and 3 3 1 2 z1 3 without actually calculating them. 4 4 z 2 5 z 6 1 6 31. Transformations 11 12 z 12 a. Describe the transformation 2 applied to the graph of the 0 1234 complex number z if z is O 13 23 multiplied by cos i sin . 12 12 b. Describe the transformation 7 11 applied to the graph of the 6 6 5 7 complex number z if z is 4 4 4 5 1 3 3 multiplied by i. 17 3 19 3 2 2 12 2 12
32. Critical Thinking Find the quadratic equation az2 bz c 0 such that a 1 5 5 and the solutions are 3 cos i sin and 2 cos i sin . 3 3 6 6
Mixed Review 33. Express 5 12i in polar form. (Lesson 9-6) 5 34. Write the equation r 5 sec in rectangular form. (Lesson 9-4) 6 35. Physics A prop for a play is supported equally by two wires suspended from the ceiling. The wires form a 130° angle with each other. If the prop weighs 23 pounds, what is the tension in each of the wires? (Lesson 8-5)
Extra Practice See p. A43. Lesson 9-7 Products and Quotients of Complex Numbers in Polar Form 597 36. Solve cos 2x sin x 1 for principal values of x. (Lesson 7-5)
37. Write the equation for the inverse of y cos x. (Lesson 6-8)
38. SAT/ACT Practice In the figure, the perimeter of square A 2 B C BCDE is how much smaller than the perimeter of rectangle ACDF? 3 A 2 B 3 C 4 FDE D 6 E 16
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598 Chapter 9 Polar Coordinates and Complex Numbers 9-8 Powers and Roots of Complex Numbers W al or OBJECTIVE e ld R COMPUTER GRAPHICS Many of the computer graphics that are • Find powers A referred to as fractals are graphs of Julia sets, which are named after and roots of p n o complex plic ati the mathematician Gaston Julia. When a function like f(z) z2 c, numbers in where c is a complex constant, is iterated, points in the complex plane can be polar form using De Moivre’s classified according to their behavior under iteration. Theorem. • Points that escape to infinity under iteration belong to the escape set of the function. • Points that do not escape belong to the prisoner set. The Julia set is the boundary between the escape set and the prisoner set. Is the number w 0.6 0.5i in the escape set or the prisoner set of the function f(z) z2? This problem will be solved in Example 6.
You can use the formula for the product of complex numbers to find the square of a complex number. [r(cos i sin )]2 [r(cos i sin )] [r(cos i sin )] r2[cos ( ) i sin ( )] r2(cos 2 i sin 2 ) Other powers of complex numbers can be found using De Moivre’s Theorem.
De Moivre’s [r(cos i sin )]n r n(cos n i sin n ) Theorem You will be asked to prove De Moivre’s Theorem in Chapter 12.
6 Example 1 Find 2 2 3 i .
First, write 2 2 3 i in polar form. Note that its graph is in the first quadrant of the complex plane. 2 r 22 2 3 Arctan 2 3 2 4 12 Arctan 3 4 3 (continued on the next page)
Lesson 9-8 Powers and Roots of Complex Numbers 599 The polar form of 2 2 3 i is 4 cos i sin . 3 3
Now use De Moivre’s Theorem to find the sixth power. 6 (2 2 3 i)6 4 cos i sin 3 3 46 cos 6 i sin 6 3 3 4096(cos 2 i sin 2 ) 4096(1 0i) Write the result in rectangular form. 4096
6 Therefore, 2 2 3 i 4096.
De Moivre’s Theorem is valid for all rational values of n. Therefore, it is also useful for finding negative powers of complex numbers and roots of complex numbers.
3 1 5 Find i . Example 2 2 2 1 First, write 3 i in polar form. Note that its graph is in the fourth 2 2 quadrant of the complex plane.
1 3 2 1 2 r Arctan 2 2 2 3 2 3 1 or 1 Arctan 3 or 4 4 3 6
1 The polar form of 3 i is 1 cos i sin . 2 2 6 6
Use De Moivre’s Theorem to find the negative 5th power. 1 5 5 3 i 1 cos i sin 2 2 6 6 1 5 cos ( 5) i sin ( 5) De Moivre’s Theorem 6 6 5 5 1 cos i sin Simplify. 6 6 1 3 i Write the answer in rectangular form. 2 2
600 Chapter 9 Polar Coordinates and Complex Numbers Recall that positive real numbers have two square roots and that the positive one is called the principal square root. In general, all nonzero complex numbers have p distinct pth roots. That is, they each have two square roots, three cube roots, four fourth roots, and so on. The principal pth root of a complex number is given by: 1 1 (a bi)p [r(cos i sin )]p 1 When finding a principal root, the interval r p cos i sin . p p is used.
3 Example 3 Find 8 i. 1 3 8 i (0 8i)3 a 0, b 8
1 3 Polar form; r 0 2 8 2 or 8, 8 cos i sin 2 2 2 since a 0.
1 1 1 83 cos i sin De Moivre’s Theorem 3 2 3 2 2 cos i sin 6 6 3 1 2 i or 3 i This is the principal cube root. 2 2
The following formula generates all of the pth roots of a complex number. It is based on the identities cos cos ( 2n ) and sin sin ( 2n ), where n is any integer.
The p distinct pth roots of a bi can be found by replacing n with The p Distinct 0, 1, 2, …, p 1, successively, in the following equation. 1 pth Roots of 1 p p a Complex (a bi) (r[cos ( 2n ) i sin ( 2n )]) 1 2n 2n Number r p cos i sin p p
Example 4 Find the three cube roots of 2 2i. First, write 2 2i in polar form. 2 5 r ( 2)2 ( 2) 2 or 2 2 Arctan or 2 4 5 5 2 2i 2 2 cos 2n i sin 2n n is any integer. 4 4 Now write an expression for the cube roots. 1 3 1 5 5 ( 2 2i)3 2 2 cos 2n i sin 2n 4 4
1 3 1 5 5 3 3 2n 2n 2 2 22 2 cos 4 i sin 4 3 3 1 2 2 or 2
(continued on the next page)
Lesson 9-8 Powers and Roots of Complex Numbers 601 Let n 0, 1, and 2 successively to find the cube roots.
5 5 2(0) 2(0) 4 4 Let n 0. 2 cos i sin 3 3 5 5 2 cos i sin 12 12 0.37 1.37i
5 5 2(1) 2(1) 4 4 Let n 1. 2 cos i sin 3 3
13 13 2 cos i sin 12 12 1.37 0.37i
5 5 2(2) 2(2) 4 4 Let n 2. 2 cos i sin 3 3 21 21 2 cos i sin 12 12 1 i
The cube roots of 2 2i are approximately 0.37 1.37i, 1.37 0.37i, and 1 i. These roots can be checked by multiplication.
GRAPHING CALCULATOR EXPLORATION
The p distinct pth roots of a complex number TRY THESE can be approximated using the parametric 1. Approximate the cube roots of 1. mode on a graphing calculator. For a particular complex number r(cos i sin ) and a 2. Approximate the fourth roots of i. particular value of p: 3. Approximate the fifth roots of 1 i. ➧ Select the Radian and Par modes. ➧ Select the viewing window. WHAT DO YOU THINK? 2 Tmin , Tmax 2 , Tstep , 4. What geometric figure is formed when you p p p 1 1 graph the three cube roots of a complex r p r p Xmin , Xmax , Xscl 1, number? 1 1 p p Ymin r , Ymax r , and Yscl 1. 5. What geometric figure is formed when you ➧ Enter the parametric equations graph the fifth roots of a complex number? 1 1 p p X1T r cos T and Y1T r sin T. 6. Under what conditions will the complex number a bi have a root that lies on the ➧ Graph the equations. positive real axis? ➧ UseTRACE to locate the roots.
602 Chapter 9 Polar Coordinates and Complex Numbers You can also use De Moivre’s Theorem to solve some polynomial equations.
Examples 5 Solve x 5 32 0. Then graph the roots in the complex plane. The solutions to this equation are the same as those of the equation x 5 32. That means we have to find the fifth roots of 32. 32 32 0i a 32, b 0 0 32(cos 0 i sin 0) Polar form; r 32 2 02 or 32, Arctan or 0 32 Now write an expression for the fifth roots.
1 1 325 [32(cos (0 2n ) i sin (0 2n ))]5 2n 2n 2 cos i sin 5 5 Let n 0, 1, 2, 3, and 4 successively to find the fifth roots, x1, x2, x3, x4, and x5. Let n 0. x1 2(cos 0 i sin 0) 2
2 2 Let n 1. x 2 cos i sin 0.62 1.90i 2 5 5
4 4 Let n 2. x 2 cos i sin 1.62 1.18i 3 5 5
6 6 Let n 3. x 2 cos i sin 1.62 1.18i 4 5 5
8 8 Let n 4. x 2 cos i sin 0.62 1.90i 5 5 5
The solutions of x 5 32 0 are 2, 0.62 1.90i, and 1.62 1.18i.
The solutions are graphed at the right. Notice i 2 0.62 1.90i that the points are the vertices of a regular pentagon. The roots of a complex number are 1 cyclical in nature. That means, when the 1.62 1.18i roots are graphed on the complex plane, the 2 roots are equally spaced around a circle. 2 1 O 1 1.62 1.18i 1
2 0.62 1.90i
Example 6 COMPUTER GRAPHICS Refer to the application at the beginning of the W lesson. Is the number w 0.6 0.5i in the escape set or the prisoner set of al or e ld 2 R the function f(z) z ?
A p n Iterating this function requires you to square complex numbers, so you can p o lic ati use De Moivre’s Theorem. Write w in polar form. r 0.6 2 ( 0.5) 2 or about 0.78 0.5 w 0.78[cos ( 0.69) i sin ( 0.69)] Arctan or about 0.69 0.6 (continued on the next page)
Lesson 9-8 Powers and Roots of Complex Numbers 603 Now iterate the function. w1 f(w) w2 Graphing 2 Calculator (0.78[cos ( 0.69) i sin ( 0.69)]) Programs 0.782[cos 2( 0.69) i sin 2( 0.69)] De Moivre’s Theorem For a program 0.12 0.60i Use a calculator to approximate that draws the rectangular form. Julia sets, visit: www.amc. w2 f(w1) 2 glencoe.com w1 2 2 (0.78 [cos 2( 0.69) i sin 2( 0.69)]) Use the polar form of w1. 0.784[cos 4( 0.69) i sin 4( 0.69)] 0.34 0.14i
w3 f(w2) 2 w2 4 2 (0.78 [cos 4( 0.69) i sin 4( 0.69)]) Use the polar form of w2. 0.788[cos 8( 0.69) i sin 8( 0.69)] 0.10 0.09i
2 4 i The moduli of these iterates are 0.78 , 0.78 , 1 0.788, and so on. These moduli will approach 0 as the number of iterations increases. This means the graphs of the iterates approach the origin in the w3 complex plane, so w 0.6 0.5i is in the prisoner 1 O 1 set of the function. w2 w w1 1
C HECK FOR U NDERSTANDING
Communicating Read and study the lesson to answer each question. Mathematics 1. Evaluate the product (1 i) (1 i) (1 i) (1 i) (1 i) by traditional multiplication. Compare the results with the results using De Moivre’s Theorem on (1 i)5. Which method do you prefer?
2. Explain how to use De Moivre’s Theorem to find the reciprocal of a complex number in polar form.
3. Graph all the fourth roots of a complex number if a ai is one of the fourth roots. Assume a is positive.
4. You Decide Shembala says that if a 0, then (a ai)2 must be a pure imaginary number. Arturo disagrees. Who is correct? Use polar form to explain.
604 Chapter 9 Polar Coordinates and Complex Numbers Guided Practice Find each power. Express the result in rectangular form. 3 5. 3 i 6. (3 5i)4
Find each principal root. Express the result in the form a bi with a and b rounded to the nearest hundredth. 1 1 7. i 6 8. ( 2 i)3
Solve each equation. Then graph the roots in the complex plane. 9. x4 i 0 10. 2x 3 4 2i 0
11. Fractals Refer to the application at the beginning of the lesson. Is w 0.8 0.7i in the prisoner set or the escape set for the function f(z) z2? Explain.
E XERCISES
Practice Find each power. Express the result in rectangular form. 3 5 A 12. 3 cos i sin 13. 2 cos i sin 6 6 4 4 4 14. ( 2 2i)3 15. 1 3 i B 16. (3 6i)4 17. (2 3i) 2
18. Raise 2 4i to the fourth power.
Find each principal root. Express the result in the form a bi with a and b rounded to the nearest hundredth.
1 2 2 5 1 19. 32 cos i sin 20. ( 1)4 3 3
1 1 21. ( 2 i)4 22. (4 i)3
1 1 23. (2 2i)3 24. ( 1 i)4
25. Find the principal square root of i.
Solve each equation. Then graph the roots in the complex plane. 26. x3 1 0 27. x5 1 0 C 28. 2x4 128 0 29. 3x4 48 0 30. x4 (1 i) 0 31. 2x4 2 2 3 i 0
Graphing Use a graphing calculator to find all of the indicated roots. Calculator 32. fifth roots of 10 9i 33. sixth roots of 2 4i 34. eighth roots of 36 20i
1 3 Applications 35. Fractals Is the number i in the escape set or the prisoner set for the 2 4 and Problem function f(z) z2? Explain. Solving W 36. Critical Thinking Suppose w a bi is one of the 31st roots of 1. al or e ld R a. What is the maximum value of a? A p n o plic ati b. What is the maximum value of b?
www.amc.glencoe.com/self_check_quiz Lesson 9-8 Powers and Roots of Complex Numbers 605 37. Design Gloribel works for an advertising agency. She wants to incorporate a hexagon design into the artwork for one of her proposals. She knows that she can locate the vertices of a regular hexagon by graphing the solutions to the equation x6 1 0 in the complex plane. What are the solutions to this equation? 38. Computer Graphics Computer programmers can use complex numbers and the complex plane to implement geometric transformations. If a programmer starts with a square with vertices at (2, 2), ( 2, 2), ( 2, 2), and (2, 2), each of the vertices can be stored as a complex number in polar form. Complex number multiplication can be used to rotate the square 45° counterclockwise and dilate it so that the new vertices lie at the midpoints of the sides of the original square. a. What complex number should the programmer multiply by to produce this transformation? b. What happens if the original vertices are multiplied by the square of your answer to part a? 39. Critical Thinking Explain why the sum of the imaginary parts of the p distinct pth roots of any positive real number must be zero.
5 5 40. Find the product 2 cos i sin 3 cos i sin . Express the result in Mixed Review 6 6 3 3 rectangular form. (Lesson 9-7) 41. Simplify (2 5i) ( 3 6i) ( 6 2i). (Lesson 9-5) 42. Write parametric equations of the line with equation y 2x 7. (Lesson 8-6) 43. Use a half-angle identity to find the exact value of cos 22.5°. (Lesson 7-4) B 44. Solve triangle ABC if A 81°15 and b 28. Round c a angle measures to the nearest minute and side measures to the nearest tenth. (Lesson 5-4) A b C
45. Manufacturing The Precious Animal Company must produce at least 300 large stuffed bears and 400 small stuffed bears per day. At most, the company can produce a total of 1200 bears per day. The profit for each large bear is $9.00, and the profit for each small bear is $5.00. How many of each type of bear should be produced each day to maximize profit? (Lesson 2-7)
46. SAT/ACT Practice Six quarts of a 20% solution of alcohol in water are mixed with 4 quarts of a 60% solution of alcohol in water. The alcoholic strength of the mixture is A 36% B 40% C 48% D 60% E 80%
606 Chapter 9 Polar Coordinates and Complex Numbers Extra Practice See p. A43. CHAPTER 9 STUDY GUIDE AND ASSESSMENT VOCABULARY
absolute value of a complex imaginary part (p. 581) polar plane (p. 553) number (p. 586) iteration (p. 581) pole (p. 553) amplitude of a complex Julia set (p. 599) prisoner set (p. 599) number (p. 587) lemniscate (p. 564) pure imaginary number Argand plane (p. 586) limaçon (p. 562) (p. 581) argument of a complex modulus (p. 587) real part (p. 581) number (p. 587) polar axis (p. 553) rectangular form of a cardioid (p. 563) polar coordinates (p. 553) complex number (p. 581) complex conjugates (p. 582) polar equation (p. 556) rose (p. 563) complex number (p. 580) polar form of a complex spiral of Archimedes (p. 564) complex plane (p. 586) number (p. 588) trigonometric form of a escape set (p. 599) polar graph (p. 556) complex number (p. 588) imaginary number (p. 581)
UNDERSTANDING AND USING THE VOCABULARY
Choose the correct term to best complete each sentence. 1. The (absolute value, conjugate) of a complex number is its distance from zero in the complex plane. 2. (Complex, Polar) coordinates give the position of an object using distances and angles. 3. Points that do not escape under iteration belong to the (escape, prisoner) set. 4. The process of repeatedly applying a function to the output produced by the previous input is called (independent, iteration). 5. A complex number in the form bi where b 0 is a (pure imaginary, real) number. 6. A (cardioid, rose) is a special type of limaçon. 7. The complex number a bi, where a and b are real numbers, is in (polar, rectangular) form. 8. The (spiral of Archimedes, lemniscate) has a polar equation of the form r a . 9. The complex plane is sometimes called the (Argand, polar) plane. 10. The polar form of a complex number is given by r(cos i sin ), where r represents the (modulus, amplitude) of the complex number.
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Chapter 9 Study Guide and Assessment 607 CHAPTER 9 • STUDY GUIDE AND ASSESSMENT
SKILLS AND CONCEPTS
OBJECTIVES AND EXAMPLES REVIEW EXERCISES
Lesson 9-1 Graph points in polar coordinates. Graph each point. 5 Graph P 2, . 11. A( 3, 50°) 6 12. B(1.5, 110°) Extend the terminal side of the angle 5 measuring . Find the point P that is 13. C 2, 6 4 2 units from the pole along the ray 14. D 3, opposite the terminal side. 2 7 5 15. Name four other pairs of polar coordinates 2 2 12 12 for the point at (4, 225°). 3 3 3 4 4 5 6 6 Graph each polar equation. 11 16. r 7 12 12 17. r 2 0 1234 O 18. 80° P 3 13 23 19. 12 12 4 7 11 6 6 5 7 4 4 4 5 3 17 3 19 3 12 2 12
Lesson 9-2 Graph polar equations. Graph each polar equation. Identify the type of curve each represents. Graph r 3 3 sin . 20. r 7 cos The graph below can be made from a table of values. This graph is a cardioid. 21. r 5 105û 90û 75û 22. r 2 4 cos 120û 60û 135û 45û 23. r 6 sin 2 150û 30û 165û 15û
O 0 180û 246 8 û
195û 345û 210û 330û 225û 315û 240û 300û 255û 270û 285û
608 Chapter 9 Polar Coordinates and Complex Numbers CHAPTER 9 • STUDY GUIDE AND ASSESSMENT
OBJECTIVES AND EXAMPLES REVIEW EXERCISES
Lesson 9-3 Convert between polar and Find the rectangular coordinates of each point rectangular coordinates. with the given polar coordinates. Find the rectangular coordinates of 24. (6, 45°) 25. (2, 330°) 3 C 3, . 26. 2, 27. 1, 6 4 2 For C 3, , r 3 and . 6 6 Find the polar coordinates of each point x r cos y r sin with the given rectangular coordinates. Use 3 cos 3 sin 0 2 and r 0. 6 6 3 3 3 28. 3 , 3 29. (5, 5) 2 2 30. ( 3, 1) 31. (4, 2) The rectangular coordinates of C are 3 3 3 , , or about ( 2.6, 1.5). 2 2
Lesson 9-4 Write the polar form of a linear Write each equation in polar form. Round to equation. the nearest degree. Write 2 x 2 y 6 in polar form. 32. 2x y 3 The standard form of the equation is 33. y 3x 4 2 x 2 y 6 0. Since A 2 B 2 2 2 or 2, the normal Write each equation in rectangular form. 2 2 form is x y 3 0. 2 2 34. 3 r cos sin 3 tan cos 35. 4 r cos tan 1 2 45° or 315° The polar form is 3 r cos ( 315°).
Lesson 9-5 Add, subtract, multiply, and divide Simplify. complex numbers in rectangular form. 36. i10 i25 Simplify (2 4i)(5 2i). 37. (2 3i) (4 4i) (2 4i)(5 2i) 2(5 2i) 4i(5 2i) 38. (2 7i) ( 3 i) 10 4i 20i 8i 2 10 16i 8( 1) 39. i3(4 3i) 18 16i 40. (i 7)( i 7) 4 2i 41. 5 2i 5 i 42. 1 2 i
Chapter 9 Study Guide and Assessment 609 CHAPTER 9 • STUDY GUIDE AND ASSESSMENT
OBJECTIVES AND EXAMPLES REVIEW EXERCISES
Lesson 9-6 Convert complex numbers from Express each complex number in polar form rectangular to polar form and vice versa. 43. 2 2i 44. 1 3i Express 5 2i in polar form. 45. 1 3 i 46. 6 4i Find the modulus. 47. 4 i 48. 4 49. 2 2 50. 3i r 5 2 ( 2)2 or 29 Find the amplitude. Since 5 2i is in Graph each complex number. Then express it Quadrant IV in the complex plane, is in in rectangular form. Quadrant IV. 2 51. 2 cos i sin Arctan or about 0.38 6 6 5 5 5 52. 3 cos i sin 5 2i 29 [cos ( 0.38) i sin ( 0.38)] 3 3
Lesson 9-7 Find the product and quotient of Find each product or quotient. Express the complex numbers in polar form. result in rectangular form. Find the product of 4 cos i sin 53. 4 cos i sin 3 cos i sin 2 2 3 3 3 3 3 3 and 3 cos i sin . Then express 54. 8 cos i sin 4 cos i sin 4 4 4 4 2 2 the product in rectangular form. 55. 2(cos 2 i sin 2) 5(cos 0.5 i sin 0.5) r r r 1 2 1 2 7 7 5 5 3 56. 8 cos i sin 2 cos i sin 4(3) 6 6 3 3 2 4 12 5 57. 6 cos i sin 4 cos i sin 2 2 6 6 4 5 5 58. 2.2(cos 1.5 i sin 1.5) The product is 12 cos i sin . 4 4 4.4(cos 0.6 i sin 0.6) Now find the rectangular form of the product. 5 5 2 2 12 cos i sin 12 i 4 4 2 2 6 2 6 2 i
Lesson 9-8 Find powers and roots of complex Find each power. Express the result in numbers in polar form using De Moivre’s rectangular form. 7 Theorem. 59. (2 2i)8 60. 3 i Find ( 3 3i)4. 61. ( 1 i)4 62. ( 2 2i)3 3 3 4 ( 3 3i)4 3 2 cos i sin 4 4 Find each principal root. Express the result in the form a bi with a and b rounded to the 4 3 2 (cos 3 i sin 3 ) nearest hundredth. 1 1 3 324( 1 0) or 324 63. i4 64. 3 i
610 Chapter 9 Polar Coordinates and Complex Numbers CHAPTER 9 • STUDY GUIDE AND ASSESSMENT
APPLICATIONS AND PROBLEM SOLVING
65. Chemistry An electron moves about the 66. Surveying A surveyor identifies a nucleus of an atom at such a high speed that landmark at the point with rectangular if it were visible to the eye, it would appear coordinates (75, 125). What are the polar as a cloud. Identify the classical curve coordinates of this point? (Lesson 9-3) represented by the electron cloud below. (Lesson 9-2) 67. Navigation A submarine sonar is tracking z a ship. The path of the ship is being coded as the equation r cos 5 0. Find x 2 the rectangular equation of the path of the ship. (Lesson 9-4) y 68. Electricity Find the current in a circuit with a voltage of 50 180j volts and an impedance of 4 5j ohms. (Lesson 9-7)
ALTERNATIVE ASSESSMENT
L OPEN-ENDED ASSESSMENT W D
W Unit 3 Project
1. The simplest form of the sum of two B I D E complex numbers is 7 4i. E W SPACE—THE FINAL FRONTIER a. Give examples of two complex numbers Coordinates in Space—What do scientists use? a b in which 0 and 0 with this sum. • Search the Internet to find types of coordinate b. Are the two complex numbers you chose systems that are used to locate objects in in part a the only two with this sum? space. Find at least two different types. Explain. • Write a summary that describes each 2. The absolute value of a complex number coordinate system that you found. Compare them to rectangular and polar coordinates. is 17 . Include diagrams illustrating how to use each a. Give a complex number in which a 0 coordinate system and any information you and b 0 with this absolute value. found about converting between systems. b. Is the complex number you chose in part a the only one with this absolute value? Explain. PORTFOLIO Choose one of the classical curves you studied in this chapter. Give the possible general forms of the polar equation for your curve and Additional Assessment See p. A64 for Chapter 9 sketch the general graph. Then write and graph practice test. a specific polar equation.
Chapter 9 Study Guide and Assessment 611 CHAPTER 9 SAT & ACT Preparation
Geometry Problems—
Lines, Angles, and Arcs TEST-TAKING TIP SAT and ACT geometry problems often combine triangles and Use the square corner of a quadrilaterals with angles and parallel lines. Problems that deal with sheet of paper to estimate circles often include arcs and angles. angle measure. Fold the corner in half to form a Review these concepts. 45° angle. • Angles: vertical, supplementary, complementary • Parallel Lines: transversal, alternate interior angles • Circles: inscribed angle, central angle, arc length, tangent line
SAT EXAMPLE ACT EXAMPLE 1. If four lines intersect as shown in the figure, 2. In the figure below, ABCD is a square x y inscribed in the circle centered at O. If OB is 6 units long, how many units long is minor 1 2 BC 135 x arc ? û û 3 A B 70 û 6 yû 4 O
A 65 B 110 C 155 D 205 D C 3 E It cannot be determined from the A B 3 C 6 D 12 E 36 information given. 2 HINT Consider all of the information given and HINT Do not assume anything from a figure. In implied. For example, a square’s diagonals this figure, and look parallel, but that 3 4 are perpendicular. information is not given. Solution Figure ABCD is a square, so its angles Solution Notice that the lines intersect to form are each 90°, and its diagonals are perpendicular. a quadrilateral. The sum of the measures of the So BOC 90°. This means that minor arc BC is interior angles of a quadrilateral is 360°. Use one-fourth of the circle. vertical angles to determine the angles in the quadrilateral. Look at the answer choices. They all include . Use the formula for the circumference of a 1 2 circle. 135 x û û 135 xû 3 C 2 r û y 2 (6) or 12 û 70û yû 4 The length of minor arc BC is one-fourth of the circumference of the circle. 1 1 C (12 ) or 3 Write an equation for the sum of the angles. 4 4 135 x 70 y 360 The answer is choice B. 205 x y 360 x y 155 The answer is choice C.
612 Chapter 9 Polar Coordinates and Complex Numbers SAT AND ACT PRACTICE
After you work each problem, record your 6. In the figure, what is answer on the answer sheet provided or on a the sum of the degree piece of paper. measures of the marked Multiple Choice angles? 1. In the figure below, line L is parallel to line M. A 180 B 270 C 360 D 540 Line N intersects both L and M, with angles a, E It cannot be determined from the b, c, d, e, f, g, and h as shown. Which of the information given. following lists includes all of the angles that are supplementary to a? 7. If 5x2 6x 70 and 5x2 6y 10, then N a b what is the value of 10x 10y? d c L e f A 10 B 20 C 60 D 80 E 100 M h g
A b, d, f, h B c, e, g C b, d, c 8. In the figure below, if A B C D , then what is the value of y? Figure not drawn to scale. D e, f, g, h E d, c, h, g C 2. In the figure below, what is the area of ABC A 120 in terms of x? x û x y û û B û D
8 B
x A 30 B 60 C 90 D 120 E 150 A û 10 C A 10 sin x B 40 sin x C 80 sin x 9. Which pair must be equal? D 40 cos x E 80 cos x M P 3. If PQRS is a parallelogram and MN is a line hû iû segment, then x must equal g jû û kû M N O c û Q P A h and i xû bû B (g h) and (i j) C (g i) and (h j) a SRû D g and j E (g j) and (h i) N A b B c C a b 180 180 10. Grid-In If 1 is parallel to 2 in the figure D a c E b c below, what is the value of y?
4. If a rectangular swimming pool has a volume 1 of 16,500 cubic feet, a depth of 10 feet, and a 110û yû length of 75 feet, what is the width of the x ûx pool, in feet? û 2 A 22 B 26 C 32 D 110 E 1650 1 1 5. 100 99 10 10 SAT/ACT Practice For additional test 9 1 1 1 9 A B C D E practice questions, visit: www.amc.glencoe.com 10100 10100 10100 10 10
SAT & ACT Preparation 613